\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2002(2002), No. 64, pp. 1--25.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2002 Southwest Texas State University.} 
\vspace{1cm}}

\begin{document} 

\title[\hfilneg EJDE--2001/64\hfil 
Expansions of solutions of evolution equations]
{Expansions of solutions of higher order evolution equations in series of \\
generalized heat polynomials} 

\author[G. N. Hile \& A. Stanoyevitch\hfil EJDE--2001/64\hfilneg]
{G. N. Hile \&  Alexander Stanoyevitch}


\address{G. N. Hile \hfill\break
Mathematics Department, University of Hawaii \hfill\break
Honolulu, HI 96822, USA}
\email{hile@math.hawaii.edu}

\address{Alexander Stanoyevitch \hfill\break
Department of Mathematics, University of Guam \hfill\break
UOG Station, Mangilao, GU 96923, USA}
\email{alex@math.hawaii.edu}

\subjclass{Primary 35C10, 35K25; Secondary 35C05, 35K30}

\keywords{heat polynomials, polynomial solutions, evolution equations, series 
expansions}

\date{}
\thanks{Submitted January 22, 2002. Published July 11, 2002.}

\subjclass[2000]{35C10, 35K25, 35C05, 35K30}
\keywords{heat polynomials, polynomial solutions, evolution equations, 
\hfill\break\indent
series expansions}


\begin{abstract}
  Upper bound estimates are established on generalized heat
  polynomials for higher order linear homogeneous evolution 
  equations with coefficients depending on the time variable. 
  These estimates are analogous to well known bounds of Rosenbloom 
  and Widder on the heat polynomials. The bounds lead to further
  estimates on the width of the strip of convergence of series
  expansions in terms of these polynomial solutions. An application
  is given to a Cauchy problem, wherein the solution is expressed 
  as the sum of a series of polynomial solutions.
\end{abstract}

\maketitle

\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}

\section{Introduction}

We consider a linear differential operator $\mathcal{L}$, represented by
\begin{equation}
\mathcal{L}u\left(  x,t\right)  =\partial_{t}u\left(  x,t\right)
-\sum_{\alpha}a_{\alpha}(t)\partial_{x}^{\alpha}u\left(  x,t\right)  \,.
\label{intop}%
\end{equation}
Here $x\in\mathbb{R}^{n}$, $t\in\mathbb{R}$, and the coefficients $\left\{
a_{\alpha}\right\}  $, of which there are only a finite number, are real
valued continuous functions of $t$ on an interval $\mathbb{I}$ containing the
origin. In an earlier paper \cite{HS} we developed explicit formulas for real
valued solutions $\left\{  p_{\beta}\right\}  $ of the initial value problems
\begin{equation}
\mathcal{L}p_{\beta}(x,t)=0\,, \quad p_{\beta}(x,0)=x^{\beta
}\,. \label{initial}%
\end{equation}
Each $p_{\beta}$ is for fixed $t$ a polynomial in $x$ of degree $\left|
\beta\right|  $, of the form
\[
p_{\beta}(x,t)=\sum_{\nu\leq\beta}c_{\nu}(t)x^{\nu}\,,
\]
where each coefficient $c_{\nu}$ is a real valued function in $C^{1}\left(
\mathbb{I}\right)  $. Moreover, when the coefficients $\left\{  a_{\alpha
}\right\}  $ are constant and $\mathcal{L}$ has no zero order term, $p_{\beta
}$ is a polynomial in both $x$ and $t$. The polynomials $\left\{  p_{\beta
}\right\}  $ are direct analogs of the classical \textit{heat polynomials},
appearing in Appell's work \cite{AP}, and studied in depth by Rosenbloom and
Widder \cite{RW,WID1,WID2,WID3}. Indeed, when $\mathcal{L}$ is the heat
operator $\mathcal{H}=\partial_{t}-\Delta$, these polynomials become the heat polynomials.

As Rosenbloom and Widder demonstrated, heat polynomials serve as a basis for
expansion of other solutions of the heat equation. In this paper we study
series expansions of the form
\begin{equation}
u\left(  x,t\right)  =\sum_{\beta}\frac{c_{\beta}}{\beta!}p_{\beta
}(x,t)\,, \label{intexp}%
\end{equation}
investigating convergence questions in a horizontal strip $\left\{  \left(
x,t\right)  :\left|  t\right|  <s\right\}  $. We discuss also the Cauchy
problem in such a strip,
\begin{equation}
\mathcal{L}u(x,t)=0\,,\;\;\;\;u\left(  x,0\right)  =f(x)\,,
\label{cauch}%
\end{equation}
and give conditions on $f$ ensuring that a solution can be found as a series
(\ref{intexp}).

At the expense of some notational complexity, we have endeavored to make our
operator $\mathcal{L}$ of (\ref{intop}) as general as possible, and we have
presupposed an arbitrary number $n$ of space dimensions. Allowing the
coefficients of $\mathcal{L}$ to depend on $t$ is significant, in that it
allows for more applications to evolution operators in the physical sciences,
where the convenience of constant coefficients is perhaps somewhat rare.
Likewise, allowing more than only one space dimension permits a much wider
range of applications. Although space dimensions higher than three do not
appear in physical situations, we allow an arbitrary space dimension $n$
because it permits treatment of the cases $n=1,2,3$ all at once, and there is
little additional burden in viewing $n$ as an integer possibly larger than
$3$. We point out that allowing dependence of the coefficients of
$\mathcal{L}$ on the space variable $x$ would be unrealistic, as in that case
polynomial solutions of the equation $\mathcal{L}u=0$ would likely be scarce.

The main purpose of this paper is the derivation of pointwise upper bounds on
the polynomials $\left\{  p_{\beta}\right\}  $, as such bounds are necessary
for meaningful applications. For example, in \cite{WID3} Widder showed, using
bounds on the heat polynomials, that a temperature function can be expanded in
a series of heat polynomials in the widest strip where it enjoys the Huygens
property. Colton in \cite{COL} used these same bounds to demonstrate that the
heat polynomials form a complete family of solutions of the heat equation in
certain domains. In \cite{HM}, Hile and Mawata used heat polynomial bounds to
help describe the behaviour of the heat operator as a mapping between weighted
Sobolev spaces in $\mathbb{R}^{n+1}$. We expect that perhaps the bounds we
establish here will be crucial for similar applications involving more general
evolution operators than the heat operator.

Rosenbloom and Widder established their bounds for the heat polynomials
$\left\{  v_{\beta}\right\}  $ with use of the Poisson representation
\[
v_{\beta}\left(  x,t\right)  =\int_{\mathbb{R}^{n}}k\left(  x-y,t\right)
\;y^{\beta}\,dy\,,
\]
where $k\left(  x,t\right)  $ is the heat kernel. However, for our more
general operators (\ref{intop}) -- which we do not assume are even parabolic
-- we have no analogous representation. Instead we borrow liberally from
techniques of H. Kemnitz \cite{KEM}, who studied polynomial expansions of
solutions of the equation
\[
\frac{\partial u\left(  x,t\right)  }{\partial t}=\frac{\partial^{r}u\left(
x,t\right)  }{\partial x^{r}}\,,
\]
where $r$ is an integer larger than $1$. For this equation Kemnitz presented
analogs of the heat polynomials,
\[
v_{r,n}\left(  x,t\right)  =n!\sum_{k+r\ell=n}\frac{x^{k}t^{\ell}}{k!\ell
!}=n!\sum_{\ell=0}^{\left[  n/r\right]  }\frac{x^{n-r\ell}t^{\ell}}{\left(
n-r\ell\right)  !\ell!}\,,
\]
and bounded these polynomials directly with use of estimates involving
factorials of integers. For our more general polynomials we strengthen and
extend Kemnitz's estimates to factorials of general nonnegative numbers, and
introduce a few additional factorial estimates of our own.

After deriving the polynomial estimates, we follow arguments of Kemnitz in
analyzing series expansions in these polynomials. We give conditions on the
coefficients $\left\{  c_{\beta}\right\}  $ in (\ref{intexp}) that guarantee a
minimum width $s$ of a strip of convergence of the series.

Again following the example of Kemnitz \cite{KEM}, we solve the Cauchy problem
(\ref{cauch}) under the assumption that $f$ has a power series expansion
\[
f\left(  x\right)  =\sum_{\beta}\frac{c_{\beta}}{\beta!}x^{\beta}%
\]
valid in all of $\mathbb{R}^{n}$. We must specify conditions on the
coefficients $\left\{  c_{\beta}\right\}  $ that guarantee convergence of the
sum (\ref{intexp}) to a solution $u$ in a horizontal strip. These conditions
at the same time mandate that $f$ can be extended as an entire function to all
of $\mathbb{C}^{n}$ obeying a certain exponential growth restriction at
infinity. Indeed, these investigations lead us to a more general discussion on
characterizations of the growth and type of an entire function of several
complex variables by conditions on the coefficients in its power series expansion.

For further work regarding generalizations of the heat polynomials, see
\cite{BRA,CH,FIT,HAI0,HAI1,HAI2,HAI3,HAI4,HAI5,HAI6,HM1,HM2,LO}.

There is an interesting body of work on the problem of determining all
polynomial solutions of systems of partial differential equations with
constant coefficients, as well as the dimensions of solution spaces of
polynomials of specified degree. For important papers and further
bibliographical references, see \cite{KAR1,KAR2,KAR3,PED1,PED2,PED3,SM}.

Another interesting application of the heat polynomials appears in the work of
L. R. Bragg and J. W. Dettman concerning transmutation operators. (See
\cite{BD2} for a list of references.) For example, in \cite{BD1} these authors
demonstrate that various polynomial solutions of elliptic and hyperbolic
equations can be obtained from the heat polynomials, and that these in turn
can be used to represent solutions of problems involving these equations.

\section{The Polynomials}

We review briefly the description of the polynomial solutions $\left\{
p_{\beta}\right\}  $ of (\ref{initial}), as presented in \cite{HS}. First some
explanation of the notation is required. Given $x=\left(  x_{1},x_{2}%
,\cdots,x_{n}\right)  \in\mathbb{R}^{n}$ and a multi-index $\alpha=\left(
\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\right)  $ in $\mathbb{R}^{n}$ we adopt
the usual notation $\alpha!=\alpha_{1}!\alpha_{2}!\cdots\alpha_{n}!$, and
\[
x^{\alpha}=x_{1}{}^{\alpha_{1}}x_{2}{}^{\alpha_{2}}\cdots x_{n}{}^{\alpha_{n}%
}\,,\;\;\;\;\;\;\partial_{x}^{\alpha}=\frac{\partial^{\left|
\alpha\right|  }}{\partial x_{1}^{\alpha_{1}}\partial x_{2}^{\alpha_{2}}%
\cdots\partial x_{n}^{\alpha_{n}}}\,.
\]
For multi-indices $\alpha$ and $\beta$ of the same dimension, we write
$\alpha\leq\beta$ whenever $\alpha_{i}\leq\beta_{i}$ for all $i$. As we assume
each $a_{\alpha}$ appearing in (\ref{intop}) is continuous on an interval
$\mathbb{I}$ containing the origin, for such $a_{\alpha}$ we may introduce the
antiderivative
\begin{equation}
b_{\alpha}\left(  t\right)  =\int_{0}^{t}a_{\alpha}\left(  s\right)
\,ds\,,\;t\in\mathbb{I}\,. \label{defineb}%
\end{equation}
We let $K$ denote the number of indices $\alpha$ appearing in (\ref{intop}),
and we label these indices as $\alpha^{1}$, $\alpha^{2}$, $\cdots$,
$\alpha^{K}$, and the corresponding coefficients as $a_{1}$, $a_{2}$, $\cdots
$, $a_{K}$. We further introduce vector functions
\[
a=\left(  a_{1},a_{2},\cdots,a_{K}\right)  \,,\;\;\;\;b=\left(
b_{1},b_{2},\cdots,b_{K}\right)  \,,%
\]
as well as a \textit{vector of multi-indices}
\[
\overline{\alpha}=\left(  \alpha^{1},\alpha^{2},\cdots,\alpha^{K}\right)
\,.
\]
Finally, given a multi-index $\sigma=\left(  \sigma_{1},\sigma_{2}%
,\cdots,\sigma_{K}\right)  $ in $\mathbb{R}^{K}$, the ``dot product''
$\overline{\alpha}\cdot\sigma$ is the $n$-dimensional multi-index
\[
\overline{\alpha}\cdot\sigma=\alpha^{1}\sigma_{1}+\alpha^{2}\sigma_{2}%
+\cdots+\alpha^{K}\sigma_{K}\;\;\;\text{.}%
\]
At times it is convenient to write this sum also as
\[
\overline{\alpha}\cdot\sigma=\sum_{\alpha}\alpha\sigma_{\alpha}\,,
\]
where $\sigma_{\alpha}=\sigma_{i}$ if $\alpha=\alpha^{i}$. The polynomials
$\left\{  p_{\beta}\right\}  $ are then defined on $\mathbb{R}^{n}%
\times\mathbb{I}$ according to
\begin{equation}
p_{\beta}(x,t):=\beta!\sum_{\gamma,\sigma:\gamma+\overline{\alpha}\cdot
\sigma=\beta}\frac{x^{\gamma}b(t)^{\sigma}}{\gamma!\sigma!}\,,
\label{poly1}%
\end{equation}
where the summation is over all multi-indices $\gamma$ in $\mathbb{R}^{n}$ and
$\sigma$ in $\mathbb{R}^{K}$ having the property that $\gamma+\overline
{\alpha}\cdot\sigma=\beta$. An alternative formulation is
\begin{equation}
p_{\beta}(x,t):=\beta!\sum_{\overline{\alpha}\cdot\sigma\leq\beta
}\frac{x^{\beta-\overline{\alpha}\cdot\sigma}b(t)^{\sigma}}{\left(
\beta-\overline{\alpha}\cdot\sigma\right)  !\sigma!}\,, \label{poly2}%
\end{equation}
where now the sum is over all multi-indices $\sigma$ in $\mathbb{R}^{K}$ such
that $\overline{\alpha}\cdot\sigma\leq\beta$.

If the operator $\mathcal{L}$ of (\ref{intop}) has no zero order term then the
summations (\ref{poly1}) and (\ref{poly2}) extend over only a finite number of
terms, and trivially each $p_{\beta}$ is as smooth as the individual terms in
the sum. Since each $b_{\alpha}$ is of class $C^{1}$ on $\mathbb{I}$, and
powers of $x$ are $C^{\infty}$ on $\mathbb{R}^{n}$, in this situation each
$p_{\beta}$, along with all its partial derivatives involving at most one
derivative with respect to $t$, is continuous in $\mathbb{R}^{n}%
\times\mathbb{I}$.

If the operator $\mathcal{L}$ of (\ref{intop}) has a zero order term, then the
summations (\ref{poly1}) and (\ref{poly2}) are over an infinite number of
terms, as now $\alpha=\left(  0,0,\cdots,0\right)  $ appears as an index in
(\ref{intop}) with corresponding nonzero $a_{\alpha}$, resulting in an
infinite number of multi-indices $\sigma$ in $\mathbb{R}^{K}$ such that
$\overline{\alpha}\cdot\sigma\leq\beta$. However it is shown in \cite{HS} that
in this event the series (\ref{poly1}) and (\ref{poly2}) are uniformly and
absolutely convergent on compact subsets of $\mathbb{R}^{n}\times\mathbb{I}$,
as are the differentiated series $\partial_{t}p_{\beta}$ and $\partial
_{x}^{\tau}p_{\beta}$ for all multi-indices $\tau$ in $\mathbb{R}^{n}$;
consequently, each $p_{\beta}$ and its derivatives $\partial_{t}p_{\beta}$ and
$\partial_{x}^{\tau}p_{\beta}$ are continuous in $\mathbb{R}^{n}%
\times\mathbb{I}$. It is demonstrated in \cite{HS} also that%
\begin{equation}
\partial_{x}^{\alpha}p_{\beta}(x,t)=\left\{
\begin{array}
[c]{cc}%
\frac{\beta!}{\left(  \beta-\alpha\right)  !}p_{\beta-\alpha}\left(
x,t\right)  \,, & \;\;\text{if }\beta\geq\alpha\,,\\
0\,, & \;\;\text{otherwise,}%
\end{array}
\right.  \label{oppol}%
\end{equation}%

\begin{equation}
\partial_{t}p_{\beta}(x,t)=\beta!\sum_{\alpha\leq\beta}\frac{a_{\alpha}%
(t)}{\left(  \beta-\alpha\right)  !}p_{\beta-\alpha}(x,t)\,, \label{oppoly}%
\end{equation}
with the latter summation over only those $\alpha$ appearing in (\ref{intop})
for which $\alpha\leq\beta$. From these formulas it follows further that all
partial derivatives of $p_{\beta}$ involving at most one derivative with
respect to $t$ and space derivatives of any order are continuous in
$\mathbb{R}^{n}\times\mathbb{I}$.

Because there are only a finite number of powers $x^{\gamma}$ with $\gamma
\leq\beta$, the functions $p_{\beta}$ are polynomials in $x$ for each fixed
$t$.

Finally, if $\mathcal{L}$ has only constant coefficients $\left\{  a_{\alpha
}\right\}  $, then from (\ref{defineb}) we have $b_{\alpha}\left(  t\right)
=ta_{\alpha}$ for each $\alpha$, and formulas (\ref{poly1}) and (\ref{poly2})
simplify to
\begin{equation}
p_{\beta}\left(  x,t\right)  =\beta!\sum_{\gamma,\sigma:\gamma+\overline
{\alpha}\cdot\sigma=\beta}\frac{a^{\sigma}x^{\gamma}t^{\left|  \sigma\right|
}}{\gamma!\sigma!}=\beta!\sum_{\overline{\alpha}\cdot\sigma\leq\beta
}\frac{a^{\sigma}x^{\beta-\overline{\alpha}\cdot\sigma}t^{\left|
\sigma\right|  }}{\left(  \beta-\overline{\alpha}\cdot\sigma\right)  !\sigma
!}\,. \label{poly3}%
\end{equation}
Again, if furthermore $\mathcal{L}$ has no zero order term then the sums are
over a finite number of terms, and in this event $p_{\beta}$ is a polynomial
in both variables $\left(  x,t\right)  $, of degree $\left|  \beta\right|  $.
(This is the situation in the case of the heat equation, for example.)

The reader is referred to \cite{HS} for further discussion of properties of
the functions $\left\{  p_{\beta}\right\}  $.

\section{Factorial Estimates}

Recall that the factorial function, originally defined for nonnegative
integers, can be extended as a $C^{\infty}$ function to the interval $\left(
-1,\infty\right)  $ by the relation
\begin{equation}
x!=\Gamma\left(  x+1\right)  \,. \label{deffact}%
\end{equation}
From properties of the gamma function it follows that $x!$ is positive on
$\left(  -1,\infty\right)  $, and increases to $+\infty$ as $x$ approaches $-1
$ from the right. Formula (\ref{deffact}) in fact defines the factorial
function on all complex numbers except the negative integers, where
$\Gamma\left(  x+1\right)  $ has poles, but we are interested mainly in the
behaviour of $x!$ on the real interval $[0,\infty)$.

In this paper we must use two different norms on vectors in $\mathbb{R}^{n}$,
but to conform with standard notation we are obliged to use the same notation
for each. For general vectors $x=\left(  x_{1},x_{2},\cdots,x_{n}\right)  $ in
$\mathbb{R}^{n}$ we use the Euclidean norm,
\begin{equation}
\left|  x\right|  =\left(  x_{1}{}^{2}+x_{2}{}^{2}+\cdots+x_{n}{}^{2}\right)
^{1/2}\,, \label{euclidnorm}%
\end{equation}
while for vectors $\alpha=\left(  \alpha_{1},\alpha_{2},\cdots,\alpha
_{n}\right)  $ in $\mathbb{R}^{n}$, all of whose components are nonnegative,
we use also the $\ell^{1}$ \textit{norm} (or \textit{multi-index norm }when
each entry of $\alpha$ is a nonnegative integer),%
\begin{equation}
\left|  \alpha\right|  =\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}\,.
\label{multinorm}%
\end{equation}
We write $\alpha\geq0$ to denote that $\alpha_{i}\geq0$ for each $i$, and in
this event say that $\alpha$ is a \textit{nonnegative vector}. To avoid
confusion between the two norms, we will use lower case Greek letters for
nonnegative vectors on which we intend to use the multi-index norm, and lower
case English letters for general vectors in $\mathbb{R}^{n}$ on which we use
the Euclidean norm.

For $x,y\in\mathbb{R}^{n}$ we write $x\leq y$, or $y\geq x$, whenever
$x_{i}\leq y_{i}$ for each $i$.

We will frequently encounter powers such as
\[
x^{\alpha}=x_{1}{}^{\alpha_{1}}x_{2}{}^{\alpha_{2}}\cdots x_{n}{}^{\alpha_{n}%
}\,,
\]
where $\alpha\geq0$. If each $\alpha_{i}$ is an integer then $x^{\alpha}$ is
defined on all $x$ in $\mathbb{R}^{n}$, but for general nonnegative $\alpha$
it is required that also $x$ be nonnegative.

Also for nonnegative vectors $\alpha$ in $\mathbb{R}^{n}$, we define $\alpha!$
as
\begin{equation}
\alpha!=\alpha_{1}!\alpha_{2}!\cdots\alpha_{n}!=\Gamma\left(  \alpha
_{1}+1\right)  \Gamma\left(  \alpha_{2}+1\right)  \cdots\Gamma\left(
\alpha_{n}+1\right)  \,. \label{deffactmore}%
\end{equation}

In this section we verify in the form of lemmas preliminary inequalities
involving factorials. Perhaps a few of these might be of some interest
independently of this paper.

\begin{lemma}
\label{facineq}For real numbers $x,y\geq0$,
\begin{equation}
x!y!\leq\left(  x+y\right)  !\,, \label{fac1}%
\end{equation}
and for nonnegative vectors $\alpha$ in $\mathbb{R}^{n}$,
\begin{equation}
\alpha!\leq\left|  \alpha\right|  !=\left(  \alpha_{1}+\alpha_{2}%
+\cdots+\alpha_{n}\right)  !\leq n^{\left|  \alpha\right|  }\alpha!\,.
\label{fac2}%
\end{equation}
\end{lemma}

\begin{proof}
By (\ref{deffact}), we verify (\ref{fac1}) if we show that, for $x,y\geq1$,
\[
\log\Gamma\left(  x\right)  +\log\Gamma\left(  y\right)  \leq\log\Gamma\left(
x+y-1\right)  \,.
\]
As functions of $x$ with $y$ held fixed, the two sides of this inequality
agree at $x=1$. Thus it is sufficient to confirm the differentiated
inequality,
\begin{equation}
\frac{\Gamma^{\prime}\left(  x\right)  }{\Gamma\left(  x\right)  }%
\leq\frac{\Gamma^{\prime}\left(  x+y-1\right)  }{\Gamma\left(  x+y-1\right)
}\,. \label{int}%
\end{equation}
But as is well known (\cite{ART}, chapter 2, or \cite{WW}, section 12.1), the
logarithmic derivative of the gamma function,
\[
\frac{d}{dt}\log\Gamma(t)=\frac{\Gamma^{\prime}(t)}{\Gamma(t)}\,,
\]
increases on $(0,\infty)$; thus (\ref{int}) holds in particular if $x,y\geq1$.

The left inequality of (\ref{fac2}) is a trivial equality if $n=1$, and for
$n\geq2$ it is an easy iteration of (\ref{fac1}) from $2$ factors to $n$ factors.

To prove the right inequality of (\ref{fac2}) we use a formula of Dirichlet (
\cite{WW}, section 12.5),
\begin{align*}
&  \Gamma\left(  \left|  \alpha\right|  \right)  \int_{P}f\left(  x_{1}%
+x_{2}+\cdots+x_{n}\right)  \;x_{1}{}^{\alpha_{1}-1}x_{2}{}^{\alpha_{2}%
-1}\cdots x_{n}{}^{\alpha_{n}-1}\,dx\\
&  =\Gamma\left(  \alpha_{1}\right)  \Gamma\left(  \alpha_{2}\right)
\cdots\Gamma\left(  \alpha_{n}\right)  \int_{0}^{1}f\left(  t\right)
\;t^{\left|  \alpha\right|  -1}\,dt\,,
\end{align*}
where $\alpha_{i}>0$ for each $i$, $f$ is any continuous real valued function
on $\left[  0,1\right]  $, and $P$ is the region in $\mathbb{R}^{n}$%
\[
P=\left\{  x:x_{i}\geq0\text{\ for each }i\text{,\ and }x_{1}+x_{2}%
+\cdots+x_{n}\leq1\right\}  \,.
\]
In Dirichlet's formula we take $f\equiv1$, replace each $\alpha_{i}$ with
$\alpha_{i}+1$, perform the integration on the right, and use $\Gamma\left(
t+1\right)  =t\Gamma\left(  t\right)  $ to obtain
\begin{equation}
\left(  \left|  \alpha\right|  +n\right)  !\int_{P}x^{\alpha}\,dx=\alpha
!\,, \label{dirich}%
\end{equation}
valid under our assumption that $\alpha_{i}\geq0$ for each $i$.

For each $i=1,2,\cdots,n$ we define
\[
P_{i}=\left\{  x\in P:x_{i}\leq x_{j}\text{\ },1\leq j\leq n\right\}  \,,
\]
so that
\begin{equation}
\int_{P}x^{\alpha}\,dx=\sum_{i=1}^{n}\int_{P_{i}}x^{\alpha}\,dx\,.
\label{sumofintegrals}%
\end{equation}
(The region where two or more of the $P_{i}{}^{\prime}s$ overlap is a subset
of a finite union of hyperplanes in $\mathbb{R}^{n}$ and thus has measure
zero.)\ Looking at the first integral in this sum, we observe that
\[
\int_{P_{1}}x^{\alpha}\,dx\geq\int_{P_{1}}x_{1}{}^{\left|  \alpha\right|
}\,dx=\int_{0}^{1/n}x_{1}{}^{\left|  \alpha\right|  }\;I\left(  x_{1}\right)
\,dx_{1}\,,
\]
where
\[
I\left(  x_{1}\right)  =\int_{x_{2},x_{3},\cdots,x_{n}\geq x_{1}%
\,,\;x_{2}+x_{3}+\cdots+x_{n}\leq1-x_{1}}dx_{2}\,dx_{3}\cdots dx_{n}\,.
\]
In this integral we substitute $y_{i}=x_{i}-x_{1}$ for $i=2$ to $n$, and
obtain
\[
I\left(  x_{1}\right)  =\int_{y_{2},y_{3},\cdots,y_{n}\geq0\,,\;y_{2}%
+y_{3}+\cdots+y_{n}\leq1-nx_{1}}\,dy_{2}\,dy_{3}\cdots dy_{n}=\frac{\left(
1-nx_{1}\right)  ^{n-1}}{\left(  n-1\right)  !}\,.
\]
Therefore,
\[
\int_{P_{1}}x^{\alpha}\,dx\geq\frac{1}{\left(  n-1\right)  !}\int_{0}%
^{1/n}x_{1}{}^{\left|  \alpha\right|  }\;\left(  1-nx_{1}\right)
^{n-1}\,dx_{1}\,.
\]
In the integral on the right we substitute $t=nx_{1}$ and find that
\[
\int_{P_{1}}x^{\alpha}\,dx\geq\frac{n^{-\left|  \alpha\right|  }}{n!}\int
_{0}^{1}t^{\left|  \alpha\right|  }\left(  1-t\right)  ^{n-1}%
\,dt=\frac{n^{-\left|  \alpha\right|  }}{n!}B\left(  \left|  \alpha\right|
+1,n\right)  \,,
\]
where the well-known \textit{beta function} $B$ (see \cite{ART}, chapter 2, or
\cite{LEB}, section 1.5) satisfies, for $p,q>0$,
\[
B\left(  p,q\right)  :=\int_{0}^{1}t^{p-1}\left(  1-t\right)  ^{q-1}%
\,dt=\frac{\Gamma\left(  p\right)  \Gamma\left(  q\right)  }{\Gamma\left(
p+q\right)  }\,.
\]
Thus,
\[
\int_{P_{1}}x^{\alpha}\,dx\geq\frac{n^{-\left|  \alpha\right|  }}%
{n!}\frac{\Gamma\left(  \left|  \alpha\right|  +1\right)  \Gamma\left(
n\right)  }{\Gamma\left(  \left|  \alpha\right|  +1+n\right)  }%
=\frac{n^{-\left|  \alpha\right|  }\left|  \alpha\right|  !}{n\left(  \left|
\alpha\right|  +n\right)  !}\,.
\]
By symmetry the same lower bound holds for each of the integrals in the
summation of (\ref{sumofintegrals}); thus
\[
\int_{P}x^{\alpha}\,dx\geq\sum_{i=1}^{n}\frac{n^{-\left|  \alpha\right|
}\left|  \alpha\right|  !}{n\left(  \left|  \alpha\right|  +n\right)
!}=\frac{n^{-\left|  \alpha\right|  }\left|  \alpha\right|  !}{\left(  \left|
\alpha\right|  +n\right)  !}\,.
\]
Finally, this inequality and (\ref{dirich}) give the right inequality of
(\ref{fac2}).
\end{proof}

\begin{lemma}
For real numbers $x\geq0$,
\begin{align}
\frac{\sqrt{2\pi}}{e}\left(  x+1\right)  ^{x+1/2}e^{-x}  &  \leq x!\leq\left(
x+1\right)  ^{x+1/2}e^{-x}\,,\label{stirling}\\
x^{x}e^{-x}  &  \leq x!\leq x^{x}e^{1-x}\sqrt{x+1}\,, \label{g1}%
\end{align}
and for $x,y\geq0$,%
\begin{equation}
\frac{y^{x}}{x!}\leq e^{y}\,. \label{g2}%
\end{equation}
(We interpret $0^{0}=1$ in these inequalities.)
\end{lemma}

\begin{proof}
The inequalities (\ref{stirling}) are minor variations on Stirling's formula;
we include the brief proofs because a statement of Stirling's formula in this
exact form is not easily found. In chapter 1 of \cite{LEB} and section 12.31
of \cite{WW} we find two equations of Binet for $\log\Gamma\left(  x\right)
$, $x>0$ -- namely,
\begin{equation}
\log\Gamma(x)=\left(  x-\frac{1}{2}\right)  \log x-x+1+\int_{0}^{\infty
}f\left(  t\right)  \left[  e^{-tx}-e^{-t}\right]  \,dt\,, \label{gam1}%
\end{equation}%
\begin{equation}
\log\Gamma(x)=\left(  x-\frac{1}{2}\right)  \log x-x+\frac{1}{2}\log2\pi
+\int_{0}^{\infty}f\left(  t\right)  e^{-tx}\,dt\,, \label{gam2}%
\end{equation}
where $f$ is given for $t\geq0$ as
\[
f\left(  t\right)  =\left[  \frac{1}{2}-\frac{1}{t}+\frac{1}{e^{t}-1}\right]
\frac{1}{t}=2\sum_{k=1}^{\infty}\frac{1}{t^{2}+4\pi^{2}k^{2}}\,.
\]
Moreover, $f$ is continuous, positive, and decreasing on $[0,\infty)$, with
$f\left(  0^{+}\right)  =1/12$. If $x\geq1$ then the integrand in (\ref{gam1})
is nonpositive, and we get
\[
\log\Gamma(x)\leq\left(  x-\frac{1}{2}\right)  \log x-x+1\,,
\]
and then the right inequality of (\ref{stirling}) by exponentiation. On the
other hand, if $x>0$ then the integrand of (\ref{gam2}) is positive, and thus
\[
\log\Gamma\left(  x\right)  \geq\left(  x-\frac{1}{2}\right)  \log
x-x+\frac{1}{2}\log2\pi\,,
\]
which yields the left side of (\ref{stirling}).

The right side of (\ref{g1}) follows from the right side of (\ref{stirling}),
and the observation%
\[
\left(  x+1\right)  ^{x+1/2}=\left(  \frac{x+1}{x}\right)  ^{x}x^{x}\sqrt
{x+1}\leq ex^{x}\sqrt{x+1}\,.
\]

To verify the left side of (\ref{g1}) we show that the function%
\[
h\left(  x\right)  =x!x^{-x}e^{x}%
\]
is increasing on $\left(  0,\infty\right)  $; then it follows that $h(x)\geq
h\left(  0^{+}\right)  =1$. We replace $x$ by $x+1$ in (\ref{gam2}) and
differentiate with respect to $x$ to find that
\[
\frac{\left(  x!\right)  ^{\prime}}{x!}=\log\left(  x+1\right)  -\frac{1}%
{2\left(  x+1\right)  }-\int_{0}^{\infty}tf(t)e^{-t\left(  x+1\right)
}\,dt\,.
\]
Using $0<f(t)\leq1/12$, we deduce that
\begin{align*}
\frac{\left(  x!\right)  ^{\prime}}{x!}  &  \geq\log\left(  x+1\right)
-\frac{1}{2\left(  x+1\right)  }-\frac{1}{12}\int_{0}^{\infty}te^{-t\left(
x+1\right)  }\,dt\\
&  =\log\left(  x+1\right)  -\frac{1}{2\left(  x+1\right)  }-\frac{1}%
{12\left(  x+1\right)  ^{2}}\,.
\end{align*}
If $x>0$ then $\log\left(  x+1\right)  \geq\log x+1/\left(  x+1\right)  $, and
consequently
\[
\frac{\left(  x!\right)  ^{\prime}}{x!}\geq\log x+\frac{1}{2\left(
x+1\right)  }-\frac{1}{12\left(  x+1\right)  ^{2}}>\log x\,.
\]
This inequality implies that, for $x>0$,
\[
\frac{d}{dx}\log h(x)=\frac{d}{dx}\left[  \log x!-x\log x+x\right]
=\frac{\left(  x!\right)  ^{\prime}}{x!}-\log x>0\,,
\]
and consequently $h\left(  x\right)  $ increases with $x$.

Finally, we write (\ref{g2}) as $y^{x}e^{-y}\leq x!$, and observe that
$y^{x}e^{-y}$ is maximized as a function of $y$ on $[0,\infty)$ by taking
$y=x$. Thus it is sufficient to verify the one variable inequality
$x^{x}e^{-x}\leq x!$, which is contained in (\ref{g1}).
\end{proof}

\begin{lemma}
\label{expineq}Let $x,\alpha\in\mathbb{R}^{n}$, $n\geq1$, with $\alpha\geq0$
and $1<\ell<\infty$. If either (a) each component $\alpha_{i}$ of $\alpha$ is
an integer, or (b) $x\geq0$, then
\begin{equation}
\frac{\left|  x^{\alpha}\right|  }{\alpha!}\leq\frac{1}{\left(  \left|
\alpha\right|  /\ell\right)  !}\exp\left[  \left(  n\left|  x\right|  \right)
^{\ell/\left(  \ell-1\right)  }\right]  \,, \label{mainfacineq}%
\end{equation}
where for $x$ we use the Euclidean norm (\ref{euclidnorm}) and for $\alpha$
the $\ell^{1}$ norm (\ref{multinorm}).
\end{lemma}

\begin{proof}
Under either condition (a) or (b), $x^{\alpha}$ is defined and
\[
\left|  x^{\alpha}\right|  =\left|  x_{1}{}^{\alpha_{1}}x_{2}{}^{\alpha_{2}%
}\cdots x_{n}{}^{\alpha_{n}}\right|  \leq\left|  x\right|  ^{\alpha_{1}%
}\left|  x\right|  ^{\alpha_{2}}\cdots\left|  x\right|  ^{\alpha_{n}}=\left|
x\right|  ^{\left|  \alpha\right|  }\,.
\]
We use this inequality, along with (\ref{fac2}), (\ref{fac1}), and (\ref{g2}),
to derive
\begin{align*}
\left|  x^{\alpha}\right|   &  \leq\frac{\left|  \alpha\right|  !\left|
x\right|  ^{\left|  \alpha\right|  }}{\left[  \frac{\left|  \alpha\right|
}{\ell}+\left(  \left|  \alpha\right|  -\frac{\left|  \alpha\right|  }{\ell
}\right)  \right]  !}\leq\frac{n^{\left|  \alpha\right|  }\alpha!\left|
x\right|  ^{\left|  \alpha\right|  }}{\left(  \frac{\left|  \alpha\right|
}{\ell}\right)  !\left(  \left|  \alpha\right|  -\frac{\left|  \alpha\right|
}{\ell}\right)  !}\\
&  =\frac{\alpha!}{\left(  \frac{\left|  \alpha\right|  }{\ell}\right)
!}\frac{\left[  \left(  n\left|  x\right|  \right)  ^{\ell/\left(
\ell-1\right)  }\right]  ^{\left(  \frac{\ell-1}{\ell}\right)  \left|
\alpha\right|  }}{\left(  \frac{\ell-1}{\ell}\left|  \alpha\right|  \right)
!}\leq\frac{\alpha!}{\left(  \frac{\left|  \alpha\right|  }{\ell}\right)
!}\exp\left[  \left(  n\left|  x\right|  \right)  ^{\ell/\left(
\ell-1\right)  }\right]  \,.
\end{align*}
The last inequality implies (\ref{mainfacineq}).
\end{proof}

\begin{lemma}
\label{trans}If $x,y,p$ are nonnegative real numbers, then
\[
x\leq y\Longrightarrow\frac{\left(  x+p\right)  !}{x!}\leq\frac{\left(
y+p\right)  !}{y!}\,.
\]
\end{lemma}

\begin{proof}
It is sufficient to confirm that the function
\[
f\left(  x\right)  =\frac{(x+p)!}{x!}=\frac{\Gamma\left(  x+p+1\right)
}{\Gamma\left(  x+1\right)  }%
\]
increases with $x$ on $[0,\infty)$. But%
\[
\frac{f\,^{\prime}\left(  x\right)  }{f\left(  x\right)  }=\frac{d}{dx}\log
f\left(  x\right)  =\frac{\Gamma^{\prime}\left(  x+p+1\right)  }{\Gamma\left(
x+p+1\right)  }-\frac{\Gamma^{\prime}\left(  x+1\right)  }{\Gamma\left(
x+1\right)  }\,.
\]
Again recalling that $\Gamma^{\prime}/\Gamma$ increases on $\left(
0,\infty\right)  $, we infer that $f\,^{\prime}\left(  x\right)  \geq0$.
\end{proof}

The next three lemmas yield estimates of certain sums involving factorials.

\begin{lemma}
\label{sumineqa}For $y=\left(  y_{1},y_{2},\cdots,y_{n}\right)  $ with all
$y_{i}\geq0$, and for real numbers $s,r\geq0$,
\begin{equation}
\sum_{\sigma\in\mathbb{R}^{n}\,,\;\left|  \sigma\right|  \leq s}\frac{\left|
y^{\sigma}\right|  r^{s-\left|  \sigma\right|  }}{\sigma!\left(  s-\left|
\sigma\right|  \right)  !}\leq\frac{\left(  \left|  y_{1}\right|  +\left|
y_{2}\right|  +\cdots+\left|  y_{n}\right|  +r\right)  ^{s}}{s!}\,,
\label{sumin}%
\end{equation}
where the summation is over all multi-indices $\sigma=\left(  \sigma
_{1},\sigma_{2},\cdots,\sigma_{n}\right)  $ in $\mathbb{R}^{n}$ such that
$\left|  \sigma\right|  \leq s$. (We interpret $0^{0}=1$.)
\end{lemma}

\begin{proof}
Since $\left|  y^{\sigma}\right|  \leq\left(  \left|  y_{1}\right|  ,\left|
y_{2}\right|  ,\cdots,\left|  y_{n}\right|  \right)  ^{\sigma}$, we may assume
that each $y_{i}\geq0$.

First we consider the case $n=1$, when $y$ and $\sigma$ are scalars. When
$r=0$ the right side of (\ref{sumin}) is $y^{s}/s!$, while the left side is
the same if $s$ is an integer, and $0$ if $s$ is not an integer. Thus we may
assume $r>0$. When $y=0$ both sides reduce to $r^{s}/s!$, so likewise we may
assume $y>0$. We set
\[
f(y)=\left(  y+r\right)  ^{s}\,.
\]
Let $\ell$ be the largest integer such that $\ell\leq s$. By Taylor's formula
with remainder, there exists $x$ between $0$ and $y$ such that
\begin{align*}
f(y)  &  =\sum_{\sigma=0}^{\ell}\frac{f^{(\sigma)}(0)}{\sigma!}y^{\sigma
}+\frac{f^{(\ell+1)}(x)}{\left(  \ell+1\right)  !}y^{\ell+1}\\
&  =\sum_{\sigma=0}^{\ell}\frac{f^{(\sigma)}(0)}{\sigma!}y^{\sigma
}+\frac{s(s-1)\cdots(s-\ell)(x+r)^{s-\ell-1}}{\left(  \ell+1\right)  !}%
y^{\ell+1}\\
&  \geq\sum_{\sigma=0}^{\ell}\frac{f^{(\sigma)}(0)}{\sigma!}y^{\sigma}%
=s!\sum_{\sigma\leq s}\frac{y^{\sigma}r^{s-\sigma}}{\sigma!\left(
s-\sigma\right)  !}\,.
\end{align*}
This inequality gives (\ref{sumin}) for $n=1.$

Next we assume the lemma holds for $n-1$, and consider the case of $n$. We use
the induction hypotheses for $n-1$ and the established result for $n=1$ to
derive
\begin{align*}
&  \sum_{\left|  \sigma\right|  \leq s}\frac{y^{\sigma}r^{s-\left|
\sigma\right|  }}{\sigma!\left(  s-\left|  \sigma\right|  \right)  !}\\
&  =\sum_{\sigma_{n}\leq s}\frac{y_{n}^{\sigma_{n}}}{\sigma_{n}!}\sum
_{\sigma_{1}+\cdots+\sigma_{n-1}\leq s-\sigma_{n}}\frac{\left(  y_{1}%
,\cdots,y_{n-1}\right)  ^{\left(  \sigma_{1},\cdots,\sigma_{n-1}\right)
}r^{s-\sigma_{n}-\sigma_{1}-\cdots-\sigma_{n-1}}}{\sigma_{1}!\cdots
\sigma_{n-1}!\left(  s-\sigma_{n}-\sigma_{1}-\cdots-\sigma_{n-1}\right)  !}\\
&  \leq\sum_{\sigma_{n}\leq s}\frac{y_{n}^{\sigma_{n}}}{\sigma_{n}%
!}\frac{\left(  y_{1}+y_{2}+\cdots+y_{n-1}+r\right)  ^{s-\sigma_{n}}}{\left(
s-\sigma_{n}\right)  !}\leq\frac{\left(  y_{1}+y_{2}+\cdots+y_{n}+r\right)
^{s}}{s!}\,,
\end{align*}
which is (\ref{sumin}).
\end{proof}

\begin{lemma}
\label{sumineqb}For real numbers $r,s\geq0$ and for any integers $m,n\geq1$,
\[
J(r,s,m,n):=\sum_{\sigma\in\mathbb{R}^{n}\,,\;\left|  \sigma\right|  \leq
s}\frac{r^{\left(  s-\left|  \sigma\right|  \right)  /m}}{\left(
\frac{\sigma}{m}\right)  !\left(  \frac{s-\left|  \sigma\right|  }{m}\right)
!}\leq m^{n}\frac{\left(  n+r\right)  !}{r!}\frac{\left(  n+r\right)  ^{s/m}%
}{\left(  \frac{s}{m}\right)  !}\,,
\]
where the summation is over all multi-indices $\sigma$ in $\mathbb{R}^{n}$
such that $\left|  \sigma\right|  \leq s$.
\end{lemma}

\begin{proof}
First we consider the case $n=1$, when the summation is over nonnegative
integers $\sigma$ such that $0\leq\sigma\leq s$.\ We may write any such
$\sigma$ in a unique way as
\begin{equation}
\sigma=m\tau-\rho\,, \label{zero0}%
\end{equation}
where $\tau$ and $\rho$ are nonnegative integers and $0\leq\rho<m$. (If
$\sigma=0$ we must take $\tau=\rho=0$; otherwise, $\tau\geq1$.) Note that
$0\leq\sigma\leq s$ implies
\begin{equation}
\frac{\rho}{m}\leq\tau\leq\frac{s+\rho}{m}\,. \label{zero1}%
\end{equation}
Conversely, if $\rho$ is an integer with $0\leq\rho<m$ and $\tau$ is a
nonnegative integer satisfying (\ref{zero1}), then $\sigma=m\tau-\rho$
satisfies $0\leq\sigma\leq s$. Thus there is a one-to-one correspondence, as
determined by (\ref{zero0}), between nonnegative integers $\sigma$ with
$\sigma\leq s$ and pairs of integers $\left(  \tau,\rho\right)  $ such that
$0\leq\rho<m$ and (\ref{zero1}) holds. It follows that we can write the
summation $J=J(r,s,m,1)$ as
\[
J(r,s,m,1)=\sum_{\rho=0}^{m-1}\sum_{\rho/m\leq\tau\leq\left(  s+\rho\right)
/m}\frac{r^{\left(  s+\rho\right)  /m-\tau}}{\left(  \tau-\frac{\rho}%
{m}\right)  !\left(  \frac{s+\rho}{m}-\tau\right)  !}\,.
\]
By Lemma \ref{trans}, if $\rho/m\leq\tau\leq\left(  s+\rho\right)  /m$ then
\[
\frac{1}{\left(  \tau-\frac{\rho}{m}\right)  !}=\frac{1}{\tau!}\frac{\tau
!}{\left(  \tau-\frac{\rho}{m}\right)  !}\leq\frac{1}{\tau!}\frac{\left(
\frac{s+\rho}{m}\right)  !}{\left(  \frac{s}{m}\right)  !}\;\;\;;
\]
therefore
\[
J(r,s,m,1)\leq\frac{1}{\left(  \frac{s}{m}\right)  !}\sum_{\rho=0}%
^{m-1}\left(  \frac{s+\rho}{m}\right)  !\sum_{\tau\leq\left(  s+\rho\right)
/m}\frac{1^{\tau}r^{\left(  s+\rho\right)  /m-\tau}}{\tau!\left(
\frac{s+\rho}{m}-\tau\right)  !}\;\text{\ }\;\text{.}%
\]
Applying Lemma \ref{sumineqa} with $n=1$ to the interior sum gives
\begin{align*}
J(r,s,m,1)  &  \leq\frac{1}{\left(  \frac{s}{m}\right)  !}\sum_{\rho=0}%
^{m-1}(1+r)^{\left(  s+\rho\right)  /m}\leq\frac{1}{\left(  \frac{s}%
{m}\right)  !}\sum_{\rho=0}^{m-1}(1+r)^{\left(  s+m\right)  /m}\\
&  =\frac{m\left(  1+r\right)  ^{\left(  s+m\right)  /m}}{\left(  \frac{s}%
{m}\right)  !}=m^{1}\frac{(1+r)!}{r!}\frac{\left(  1+r\right)  ^{s/m}}{\left(
\frac{s}{m}\right)  !}\,.
\end{align*}

Having established the Lemma for $n=1$, next we assume it is valid also for
$n-1$. Then
\begin{align*}
J(r,s,m,n)  &  =\sum_{\sigma\in\mathbb{R}^{n}\,,\;\left|  \sigma\right|  \leq
s}\frac{r^{\left(  s-\left|  \sigma\right|  \right)  /m}}{\left(
\frac{\sigma}{m}\right)  !\left(  \frac{s-\left|  \sigma\right|  }{m}\right)
!}\\
&  =\sum_{\sigma_{n}\leq s}\frac{1}{\left(  \frac{\sigma_{n}}{m}\right)
!}\sum_{\sigma_{1}+\cdots+\sigma_{n-1}\leq s-\sigma_{n}}\frac{r^{\left(
s-\sigma_{1}-\cdots-\sigma_{n}\right)  /m}}{\left(  \frac{\sigma_{1}}%
{m}\right)  !\cdots\left(  \frac{\sigma_{n-1}}{m}\right)  !\left(
\frac{s-\sigma_{1}-\cdots-\sigma_{n}}{m}\right)  !}\\
&  =\sum_{\sigma_{n}\leq s}\frac{1}{\left(  \frac{\sigma_{n}}{m}\right)
!}J(r,s-\sigma_{n},m,n-1)\\
&  \leq\sum_{\sigma_{n}\leq s}\frac{1}{\left(  \frac{\sigma_{n}}{m}\right)
!}m^{n-1}\frac{\left(  n-1+r\right)  !}{r!}\frac{\left(  n-1+r\right)
^{\left(  s-\sigma_{n}\right)  /m}}{\left(  \frac{s-\sigma_{n}}{m}\right)
!}\\
&  =m^{n-1}\frac{\left(  n-1+r\right)  !}{r!}J(n-1+r,s,m,1)\,.
\end{align*}

Applying the lemma to $J(n-1+r,s,m,1)$ now gives the desired inequality for
$J(r,s,m,n)$.
\end{proof}

\begin{lemma}
\label{sumineqc}Let $\mathcal{M}$ denote a nonempty collection of $M$ nonzero
multi-indices $\alpha$ in $\mathbb{R}^{n}$, and let $m$ be a positive integer
and $s$ a nonnegative real number.\ Then
\begin{equation}
\sum_{\left|  \overline{\alpha}\cdot\sigma\right|  \leq s}\frac{1}{\left(
\frac{s-\left|  \overline{\alpha}\cdot\sigma\right|  }{m}\right)
!\prod_{\alpha\in\mathcal{M}}\left(  \frac{\left|  \alpha\right|  }{m}%
\sigma_{\alpha}\right)  !}\leq m^{M}\left(  M+1\right)  !\frac{\left(
M+1\right)  ^{s/m}}{\left(  \frac{s}{m}\right)  !}\,, \label{sum1}%
\end{equation}
where the summation is taken over all multi-indices $\sigma$ in $R^{M}$ such
that
\begin{equation}
\left|  \overline{\alpha}\cdot\sigma\right|  =\left|  \sum_{\alpha
\in\mathcal{M}}\alpha\sigma_{\alpha}\right|  =\sum_{\alpha\in\mathcal{M}%
}\left|  \alpha\right|  \sigma_{\alpha}\leq s\,. \label{sum2}%
\end{equation}
\end{lemma}

\begin{proof}
Let $\ell$ denote the maximum magnitude of any multi-index in $\mathcal{M}$,
so that $\left|  \alpha\right|  \leq\ell$ for all $\alpha\in\mathcal{M}$.
First we consider the special case when $\left|  \alpha\right|  =\ell$ for all
$\alpha\in\mathcal{M}$. Then (\ref{sum2})\ shows that $\left|  \overline
{\alpha}\cdot\sigma\right|  =\ell\left|  \sigma\right|  =\left|  \ell
\sigma\right|  $. Denoting the sum on the left of (\ref{sum1}) as $S$, we have%
\[
S=\sum_{\sigma\in\mathbb{R}^{M}\,,\;\left|  \ell\sigma\right|  \leq s}%
\frac{1}{\left(  \frac{s-\left|  \ell\sigma\right|  }{m}\right)  !\left(
\frac{\ell\sigma}{m}\right)  !}\,.
\]
We can view this sum as taken over all multi-indices $\nu\in\mathbb{R}^{M}$
such that $\left|  \nu\right|  \leq s$ and $\nu$ has the special form
$\nu=\ell\sigma$ for some multi-index $\sigma\in\mathbb{R}^{M}$. The sum will
increase if we remove the last restriction on $\nu$, allowing us to use Lemma
\ref{sumineqb} to deduce that%
\[
S\leq\sum_{\nu\in\mathbb{R}^{M}\,,\;\left|  \nu\right|  \leq s}\frac{1}%
{\left(  \frac{s-\left|  \nu\right|  }{m}\right)  !\left(  \frac{\nu}%
{m}\right)  !}=J(1,s,m,M)\leq m^{M}(M+1)!\frac{\left(  M+1\right)  ^{s/m}%
}{\left(  \frac{s}{m}\right)  !}\,.
\]

We now proceed by induction on $\ell$. When $\ell=1$ we have $\left|
\alpha\right|  =1=\ell$ for all $\alpha\in\mathcal{M}$, and the result has
been established. Now we assume the lemma is valid for $1$, $2$,$\;\cdots$,
$\ell-1$, and consider the case of $\ell$. We let $\mathcal{M}_{\ell}$ consist
of all multi-indices $\alpha$ in $\mathcal{M}$ with $\left|  \alpha\right|
=\ell$, and we let $\mathcal{M}_{0}$ consist of all remaining multi-indices in
$\mathcal{M}$. We denote by $L$ and $K$ the number of multi-indices in
$\mathcal{M}_{\ell}$ and $\mathcal{M}_{0}$, respectively, so that $M=L+K$. We
have $L>0$ by assumption, and we may assume $K>0$ as otherwise the result is
already proved. Given any multi-index $\sigma$ in $\mathbb{R}^{M}$ we may
write
\[
\sigma=\tau+\lambda\,,
\]
where $\tau$ and $\lambda$ are multi-indices also in $\mathbb{R}^{M}$, defined
according to
\begin{equation}
\tau_{\alpha}=\left\{
\begin{array}
[c]{cc}%
\sigma_{\alpha}\text{\ \ ,} & \;\text{if }\left|  \alpha\right|  =\ell\,,\\
0\text{\ \ ,} & \;\text{if }0<\left|  \alpha\right|  <\ell\,,
\end{array}
\right.  \label{sum3}%
\end{equation}%
\begin{equation}
\lambda_{\alpha}=\left\{
\begin{array}
[c]{cc}%
0\text{\ \ ,} & \;\text{if }\left|  \alpha\right|  =\ell\,,\\
\sigma_{\alpha}\text{\ \ ,} & \;\text{if }0<\left|  \alpha\right|  <\ell\,.
\end{array}
\right.  \label{sum4}%
\end{equation}
Then
\begin{gather*}
\left|  \overline{\alpha}\cdot\sigma\right|  =\left|  \overline{\alpha}%
\cdot\tau\right|  +\left|  \overline{\alpha}\cdot\lambda\right|
\,,\;\;\;\;\;\left|  \overline{\alpha}\cdot\tau\right|  =\ell\left|
\tau\right|  =\left|  \ell\tau\right|  \,,\\
\prod_{\alpha\in\mathcal{M}}\left(  \frac{\left|  \alpha\right|  }{m}%
\sigma_{\alpha}\right)  !=\prod_{\alpha\in\mathcal{M}_{\ell}}\left(
\frac{\ell}{m}\tau_{\alpha}\right)  !\prod_{\alpha\in\mathcal{M}_{0}}\left(
\frac{\left|  \alpha\right|  }{m}\lambda_{\alpha}\right)  !\,,
\end{gather*}
and
\[
S=\sum_{\left|  \overline{\alpha}\cdot\tau\right|  +\left|  \overline{\alpha
}\cdot\lambda\right|  \leq s}\frac{1}{\left(  \frac{s-\left|  \overline
{\alpha}\cdot\tau\right|  -\left|  \overline{\alpha}\cdot\lambda\right|  }%
{m}\right)  !\left(  \frac{\ell\tau}{m}\right)  !\prod_{\alpha\in
\mathcal{M}_{0}}\left(  \frac{\left|  \alpha\right|  }{m}\lambda_{\alpha
}\right)  !}=\sum_{\left|  \ell\tau\right|  \leq s}\frac{1}{\left(
\frac{\ell\tau}{m}\right)  !}\cdot S_{0}\,,
\]
where%
\[
S_{0}:=\sum_{\left|  \overline{\alpha}\cdot\lambda\right|  \leq s-\left|
\ell\tau\right|  }\frac{1}{\left(  \frac{s-\left|  \ell\tau\right|  -\left|
\overline{\alpha}\cdot\lambda\right|  }{m}\right)  !\prod_{\alpha
\in\mathcal{M}_{0}}\left(  \frac{\left|  \alpha\right|  }{m}\lambda_{\alpha
}\right)  !}\,.
\]
Since%
\[
\overline{\alpha}\cdot\lambda=\sum_{\alpha\in\mathcal{M}_{0}}\alpha
\lambda_{\alpha}\,,
\]
and because the zero entries of $\lambda$ play no role in the calculations, we
may disregard the zero entries and regard $\lambda$ as a multi-index in
$\mathbb{R}^{K}$; then by the induction hypotheses,
\[
S_{0}\leq m^{K}\left(  K+1\right)  !\frac{\left(  K+1\right)  ^{\left(
s-\left|  \ell\tau\right|  \right)  /m}}{\left(  \frac{s-\left|  \ell
\tau\right|  }{m}\right)  !}\,,
\]
and consequently%
\[
S\leq m^{K}\left(  K+1\right)  !\sum_{\left|  \ell\tau\right|  \leq
s}\frac{\left(  K+1\right)  ^{\left(  s-\left|  \ell\tau\right|  \right)  /m}%
}{\left(  \frac{\ell\tau}{m}\right)  !\left(  \frac{s-\left|  \ell\tau\right|
}{m}\right)  !}\,.
\]
As the zero entries of $\tau$ play no role in the calculations, we may ignore
these entries as well and regard $\tau$ as a multi-index in $\mathbb{R}^{L}$;
then we use Lemma \ref{sumineqb} once again to derive%
\begin{align*}
S  &  \leq m^{K}\left(  K+1\right)  !\sum_{\nu\in\mathbb{R}^{L}\,,\;\left|
\nu\right|  \leq s}\frac{\left(  K+1\right)  ^{\left(  s-\left|  \nu\right|
\right)  /m}}{\left(  \frac{\nu}{m}\right)  !\left(  \frac{s-\left|
\nu\right|  }{m}\right)  !}\\
&  \leq m^{K}\left(  K+1\right)  !m^{L}\frac{\left(  K+L+1\right)  !}{\left(
K+1\right)  !}\frac{\left(  K+L+1\right)  ^{s/m}}{\left(  s/m\right)  !}\,,
\end{align*}
which with the substitution $M=K+L$ gives (\ref{sum1}).
\end{proof}

\section{Polynomial Bounds}

We now derive bounds on the polynomials $\left\{  p_{\beta}\right\}  $, as
specified by (\ref{poly1}) and (\ref{poly2}), for the operator (\ref{intop}).
We assume that the highest order of any space derivative is $m$, so that the
operator may be written as%
\begin{equation}
\mathcal{L}u\left(  x,t\right)  =\partial_{t}u\left(  x,t\right)
-\sum_{\alpha\in\mathcal{A}}a_{\alpha}(t)\partial_{x}^{\alpha}u\left(
x,t\right)  \,, \label{genop}%
\end{equation}
where $\mathcal{A}$ is a finite collection of multi-indices $\alpha$ in
$\mathbb{R}^{n}$ with $\left|  \alpha\right|  \leq m$ for each $\alpha
\in\mathcal{A}$, and with $\left|  \alpha\right|  =m$ for at least one such
$\alpha$. We suppose also that there are finite bounds $\left\{
\Lambda_{\alpha}\right\}  $ and $\Lambda$ such that%
\begin{equation}
\Lambda_{\alpha}=\sup_{t\in\mathbb{I}}\;\left|  a_{\alpha}(t)\right|
\;\;\;\left(  \alpha\in\mathcal{A}\right)  \,,\;\;\;\;\;\Lambda
=\sup_{t\in\mathbb{I}}\sum_{\alpha\in\mathcal{A}\text{\ ,\ }\left|
\alpha\right|  =m}\left|  a_{\alpha}(t)\right|  \,. \label{bnd}%
\end{equation}

The utility of Theorem \ref{genthm1} below will arise mainly from the nature
of the dependence of the bounds on $\beta$. Series in the polynomials
$\left\{  p_{\beta}\right\}  $ will, under rather general circumstances,
converge in strips in $\mathbb{R}^{n+1}$ of the form $\left|  t\right|  <s$,
and convergence will be uniform on compact subsets. Our goal in producing
inequality (\ref{gen1}) is to obtain the most efficient bounds possible in
terms of the $\beta$ -- dependent factors, so as to get the best estimate on
$s$, while being less concerned with factors independent of $\beta$. In later
results on strips of convergence we will see that the $\beta$ -- independent
bounds, as magnanimous as they seem, turn out to be easily manageable as they
are trivially bounded on compact sets in $\mathbb{R}^{n+1}$. While for any
single polynomial $p_{\beta}$ a bound of the type (\ref{gen1}) below might
seem obvious and overly generous, the point is that the inequality is valid
for all $\left\{  p_{\beta}\right\}  $ simultaneously. Note that, in our
assumptions (\ref{bnd}) on the coefficients of $\mathcal{L}$, the bound
$\Lambda$ on the highest order coefficients is singled out separately. As we
will see in later results, it is this bound that is relevant for the width of
the strip of convergence, while the bounds on the lower order coefficients are
less significant.

\begin{theorem}
\label{genthm1}Assume $m\geq2$, where $m$ is the order of (\ref{genop}), and
that the coefficents $\left\{  a_{\alpha}\right\}  $ are continuous with the
bounds (\ref{bnd}) in an interval $\mathbb{I}$ containing the origin. Then for
$x\in\mathbb{R}^{n}$ and $t\in\mathbb{I}$, and for any $\delta>0$,%
\begin{align}
\frac{\left|  p_{\beta}(x,t)\right|  }{\beta!}  &  \leq\frac{m^{M}\left(
M+1\right)  !}{\left(  \left|  \beta\right|  /m\right)  !}\left(
\Lambda\left|  t\right|  +\delta\right)  ^{\left|  \beta\right|
/m}\label{gen1}\\
&  \cdot\exp\left[  \left(  \frac{M+1}{\delta}\right)  ^{1/\left(  m-1\right)
}\left(  n\left|  x\right|  \right)  ^{m/(m-1)}\right. \nonumber\\
&  +\left.  \sum_{\alpha\in\mathcal{A}\text{\ ,\ }\left|  \alpha\right|
<m}\left(  \frac{M+1}{\delta}\right)  ^{\left|  \alpha\right|  /\left(
m-\left|  \alpha\right|  \right)  }\left|  \Lambda_{\alpha}t\right|
^{m/\left(  m-\left|  \alpha\right|  \right)  }\right]  \,.\nonumber
\end{align}
where $M$ is the number of multi-indices $\alpha\in\mathcal{A}$ such that
$0<\left|  \alpha\right|  <m$.
\end{theorem}

\begin{proof}
We initially assume (\ref{genop}) contains no zero order term; that is, there
is no term in the sum corresponding to $\alpha=\left(  0,\cdots,0\right)  $.
Let $K$ denote the number of space derivative terms in this sum; then by
(\ref{poly1}),%
\begin{equation}
\frac{p_{\beta}(x,t)}{\beta!}=\sum_{\gamma+\overline{\alpha}\cdot\sigma=\beta
}\frac{x^{\gamma}b(t)^{\sigma}}{\gamma!\sigma!}\,, \label{new1}%
\end{equation}
where the sum is over all multi-indices $\gamma$ in $\mathbb{R}^{n}$ and
$\sigma$ in $\mathbb{R}^{K}$ satisfying $\gamma+\overline{\alpha}\cdot
\sigma=\beta$. Given $\delta>0$, Lemma \ref{expineq} (with $\ell=m$) yields%
\begin{align*}
\;\;\;\frac{\left|  x^{\gamma}\right|  }{\gamma!}  &  =\frac{\delta^{\left|
\gamma\right|  /m}}{\gamma!}\left|  \left(  \frac{x_{1}}{\delta^{1/m}%
},\frac{x_{2}}{\delta^{1/m}},\cdots,\frac{x_{n}}{\delta^{1/m}}\right)
^{\gamma}\right| \\
&  \leq\frac{\delta^{\left|  \gamma\right|  /m}}{\left(  \left|
\gamma\right|  /m\right)  !}\exp\left(  \frac{n\left|  x\right|  }%
{\delta^{1/m}}\right)  ^{m/(m-1)}\,.
\end{align*}
This estimate, used in (\ref{new1}), leads to the bound%
\begin{equation}
\frac{\left|  p_{\beta}(x,t)\right|  }{\beta!}\leq\exp\left(  \frac{n\left|
x\right|  }{\delta^{1/m}}\right)  ^{m/(m-1)}\cdot S\,, \label{new2}%
\end{equation}
where%
\begin{equation}
S=\sum_{\gamma+\overline{\alpha}\cdot\sigma=\beta}\frac{\delta^{\left|
\gamma\right|  /m}}{\left(  \left|  \gamma\right|  /m\right)  !}\frac{\left|
b^{\sigma}\right|  }{\sigma!}\,. \label{new3}%
\end{equation}
(Henceforth, for brevity we suppress the arguments of functions of $t$.)

We let $N$ denote the number of terms in the summation of (\ref{genop})
corresponding to $\left|  \alpha\right|  =m$, and $M$ the number with
$0<\left|  \alpha\right|  <m$, so that $K=N+M$. We number the multi-indices
appearing in this sum as%
\[
\alpha^{1}\,,\;\;\alpha^{2}\,,\;\;\cdots\,,\;\;\alpha^{N}\,,\;\;\alpha
^{N+1}\,,\;\;\alpha^{N+2}\,,\;\;\cdots\,,\;\;\alpha^{N+M}\,,
\]
where $\alpha^{1},\cdots,\alpha^{N}$ are the highest order multi-indices with
$\left|  \alpha\right|  =m$, and \\ 
$\alpha^{N+1},\cdots,\alpha^{N+M}$ are of
lower order. (Thus $N\geq1$, as $\mathcal{L}$ has order $m$). 
We write
a general multi-index $\sigma$ in $\mathbb{R}^{K}$ as%
\[
\sigma=\left(  \nu_{1},\cdots,\nu_{N},\lambda_{1},\cdots,\lambda_{M}\right)
=\nu+\lambda\,,
\]
where the multi-indices $\nu$ and $\lambda$, each also in $\mathbb{R}^{K}$,
are defined as%
\begin{equation}
\nu=\left(  \nu_{1},\cdots,\nu_{N},0,\cdots,0\right)
\,,\;\;\;\;\;\lambda=\left(  0,\cdots,0,\lambda_{1},\cdots,\lambda
_{M}\right)  \,, \label{defs}%
\end{equation}
and%
\[
\overline{\alpha}\cdot\sigma=\sum\alpha\sigma_{\alpha}=\sum_{i=1}^{N}%
\alpha^{i}\nu_{i}+\sum_{i=1}^{M}\alpha^{N+i}\lambda_{i}\,.
\]
(We admit the possibility $M=0$, in which case $\lambda=(0,\cdots,0)$ and
$\sigma=\nu$.) Then%
\[
\overline{\alpha}\cdot\sigma=\overline{\alpha}\cdot\nu+\overline{\alpha}%
\cdot\lambda\,,\;\;\;\;\sigma!=\nu!\lambda!\,,
\]
and (\ref{new3}) can be written as%
\begin{align}
S  &  =\sum_{\gamma+\overline{\alpha}\cdot\nu+\overline{\alpha}\cdot
\lambda=\beta}\frac{\delta^{\left|  \gamma\right|  /m}}{\left(  \left|
\gamma\right|  /m\right)  !}\frac{\left|  b^{\nu}\right|  }{\nu!}\frac{\left|
b^{\lambda}\right|  }{\lambda!}\label{new4}\\
&  =\sum_{\overline{\alpha}\cdot\nu+\overline{\alpha}\cdot\lambda\leq\beta
}\frac{\delta^{\left|  \beta-\overline{\alpha}\cdot\nu-\overline{\alpha}%
\cdot\lambda\right|  /m}}{\left(  \left|  \beta-\overline{\alpha}\cdot
\nu-\overline{\alpha}\cdot\lambda\right|  /m\right)  !}\frac{\left|  b^{\nu
}\right|  }{\nu!}\frac{\left|  b^{\lambda}\right|  }{\lambda!}\nonumber\\
&  =\sum_{\overline{\alpha}\cdot\nu\leq\beta}\delta^{\left|  \beta
-\overline{\alpha}\cdot\nu\right|  /m}\frac{\left|  b^{\nu}\right|  }{\nu
!}\cdot T\,,\nonumber
\end{align}
where
\begin{align}
T  &  =\frac{1}{\left(  \left|  \beta-\overline{\alpha}\cdot\nu\right|
/m\right)  !}\,,\;\text{if }M=0\text{\ \ ,}\label{new5}\\
T  &  =\sum_{\overline{\alpha}\cdot\lambda\leq\beta-\overline{\alpha}\cdot\nu
}\frac{\delta^{-\left|  \overline{\alpha}\cdot\lambda\right|  /m}}{\left(
\left|  \beta-\overline{\alpha}\cdot\nu-\overline{\alpha}\cdot\lambda\right|
/m\right)  !}\frac{\left|  b^{\lambda}\right|  }{\lambda!}\,,\;\text{if
}M>0\text{\ .} \label{new6}%
\end{align}

In the case $M>0$ we let $\mathcal{M}$ denote the collection of multi-indices
$\left|  \alpha\right|  $ appearing in (\ref{genop}) such that $0<\left|
\alpha\right|  <m$, and apply Lemma \ref{expineq} but with $\ell=m/\left|
\alpha\right|  $ and $n=1$ to estimate%
\begin{align*}
\frac{\left|  b^{\lambda}\right|  }{\lambda!}  &  =\prod_{\alpha\in
\mathcal{M}}\frac{\left|  b_{\alpha}\right|  ^{\lambda_{\alpha}}}%
{\lambda_{\alpha}!}=\prod_{\alpha\in\mathcal{M}}\left\{  \delta^{\left|
\alpha\right|  \lambda_{\alpha}/m}\frac{1}{\lambda_{\alpha}!}\left(
\frac{\left|  b_{\alpha}\right|  }{\delta^{\left|  \alpha\right|  /m}}\right)
^{\lambda_{\alpha}}\right\} \\
&  \leq\prod_{\alpha\in\mathcal{M}}\left\{  \delta^{\left|  \alpha\right|
\lambda_{\alpha}/m}\frac{1}{\left(  \frac{\left|  \alpha\right|
\lambda_{\alpha}}{m}\right)  !}\exp\left[  \left(  \frac{\left|  b_{\alpha
}\right|  }{\delta^{\left|  \alpha\right|  /m}}\right)  ^{m/\left(  m-\left|
\alpha\right|  \right)  }\right]  \right\} \\
&  =\delta^{\left|  \overline{\alpha}\cdot\lambda\right|  /m}\exp\left[
\sum_{\alpha\in\mathcal{M}}\left(  \frac{\left|  b_{\alpha}\right|  }%
{\delta^{\left|  \alpha\right|  /m}}\right)  ^{m/\left(  m-\left|
\alpha\right|  \right)  }\right]  \prod_{\alpha\in\mathcal{M}}\frac{1}{\left(
\frac{\left|  \alpha\right|  \lambda_{\alpha}}{m}\right)  !}\,.
\end{align*}
We substitute this inequality into (\ref{new6}), noting that $\left|
\overline{\alpha}\cdot\lambda\right|  \leq\left|  \beta-\overline{\alpha}%
\cdot\nu\right|  $ in this sum, to obtain%
\begin{align*}
T  &  \leq\exp\left[  \sum_{\alpha\in\mathcal{M}}\left(  \frac{\left|
b_{\alpha}\right|  }{\delta^{\left|  \alpha\right|  /m}}\right)  ^{m/\left(
m-\left|  \alpha\right|  \right)  }\right] \\
&  \cdot\sum_{\left|  \overline{\alpha}\cdot\lambda\right|  \leq\left|
\beta-\overline{\alpha}\cdot\nu\right|  }\frac{1}{\left(  \frac{\left|
\beta-\overline{\alpha}\cdot\nu\right|  -\left|  \overline{\alpha}\cdot
\lambda\right|  }{m}\right)  !\prod_{\alpha\in\mathcal{M}}\left(
\frac{\left|  \alpha\right|  }{m}\lambda_{\alpha}\right)  !}\,.
\end{align*}
Applying Lemma \ref{sumineqc} to estimate the latter sum, we arrive at%
\[
T\leq m^{M}\left(  M+1\right)  !\frac{\left(  M+1\right)  ^{\left|
\beta-\overline{\alpha}\cdot\nu\right|  /m}}{\left(  \frac{\left|
\beta-\overline{\alpha}\cdot\nu\right|  }{m}\right)  !}\exp\left[
\sum_{\alpha\in\mathcal{M}}\left(  \frac{\left|  b_{\alpha}\right|  }%
{\delta^{\left|  \alpha\right|  /m}}\right)  ^{m/\left(  m-\left|
\alpha\right|  \right)  }\right]  \,.
\]
Note that this estimate reduces to (\ref{new5}) in the case $M=0$, when
$\mathcal{M}$ is empty, so that in fact it is valid for both $M>0$ and $M=0$.
We insert this estimate into (\ref{new4}), and then (\ref{new4}) into
(\ref{new2}), to obtain%
\begin{gather}
\frac{\left|  p_{\beta}(x,t)\right|  }{\beta!}\leq m^{M}\left(  M+1\right)
!\cdot U\label{new7}\\
\cdot\exp\left[  \left(  \frac{n\left|  x\right|  }{\delta^{1/m}}\right)
^{m/(m-1)}+\sum_{\alpha\in\mathcal{M}}\left(  \frac{\left|  b_{\alpha}\right|
}{\delta^{\left|  \alpha\right|  /m}}\right)  ^{m/\left(  m-\left|
\alpha\right|  \right)  }\right]  \,,\nonumber
\end{gather}
where%
\begin{equation}
U=\sum_{\overline{\alpha}\cdot\nu\leq\beta}\frac{\left|  b^{\nu}\right|  }%
{\nu!}\frac{\delta^{\left|  \beta-\overline{\alpha}\cdot\nu\right|  /m}\left(
M+1\right)  ^{\left|  \beta-\overline{\alpha}\cdot\nu\right|  /m}}{\left(
\left|  \beta-\overline{\alpha}\cdot\nu\right|  /m\right)  !}\,.
\label{new8}%
\end{equation}

Now, in (\ref{new8}) we have $\left|  \overline{\alpha}\cdot\nu\right|
=m\left|  \nu\right|  \leq\left|  \beta\right|  $, which allows us to use
Lemma \ref{sumineqa} to estimate%
\begin{align}
U  &  \leq\sum_{\left|  \nu\right|  \leq\left|  \beta\right|  /m}\frac{\left|
b^{\nu}\right|  }{\nu!}\frac{\left[  \delta\left(  M+1\right)  \right]
^{\left|  \beta\right|  /m-\left|  \nu\right|  }}{\left(  \left|
\beta\right|  /m-\left|  \nu\right|  \right)  !}\label{new9}\\
&  \leq\frac{1}{\left(  \left|  \beta\right|  /m\right)  !}\left[
\sum_{\left|  \alpha\right|  =m}\left|  b_{\alpha}(t)\right|  +\delta\left(
M+1\right)  \right]  ^{\left|  \beta\right|  /m}\,.\nonumber
\end{align}
We substitute this inequality into (\ref{new7}) and replace $\delta$ by
$\delta/(M+1)$ to obtain%
\begin{gather}
\frac{\left|  p_{\beta}(x,t)\right|  }{\beta!}\leq\frac{m^{M}\left(
M+1\right)  !}{\left(  \left|  \beta\right|  /m\right)  !}\left[
\sum_{\left|  \alpha\right|  =m}\left|  b_{\alpha}(t)\right|  +\delta\right]
^{\left|  \beta\right|  /m}\label{new10}\\
\cdot\exp\left[  \left(  \frac{M+1}{\delta}\right)  ^{1/\left(  m-1\right)
}\left(  n\left|  x\right|  \right)  ^{m/(m-1)}\right. \nonumber\\
\left.  +\sum_{\alpha\in\mathcal{A}\text{\ ,\ }\left|  \alpha\right|
<m}\left(  \frac{M+1}{\delta}\right)  ^{\left|  \alpha\right|  /\left(
m-\left|  \alpha\right|  \right)  }\left|  b_{\alpha}\right|  ^{m/\left(
m-\left|  \alpha\right|  \right)  }\right]  \,.\nonumber
\end{gather}
From (\ref{defineb}) and (\ref{bnd}) we deduce that%
\begin{equation}
\left|  b_{\alpha}(t)\right|  \leq\Lambda_{\alpha}\left|  t\right|
\,,\;\;\;\;\;\sum_{\left|  \alpha\right|  =m}\left|  b_{\alpha
}(t)\right|  \leq\Lambda\left|  t\right|  \,, \label{new11}%
\end{equation}
and we insert these estimates into (\ref{new10}) to arrive at (\ref{gen1}).

It remains now only to verify the theorem when the operator (\ref{genop}) has
a zero order term, corresponding to $\alpha_{0}:=\left(  0,\cdots,0\right)  $.
To analyze this situation, we assume the operator $\mathcal{L}$ of
(\ref{genop}) has no zero order term, and we subtract such a term from
$\mathcal{L}$ to produce a modified operator%
\begin{align*}
\mathcal{L}_{0}u\left(  x,t\right)   &  =\partial_{t}u\left(  x,t\right)
-\sum_{\alpha\in\mathcal{A}}a_{\alpha}(t)\partial_{x}^{\alpha}u\left(
x,t\right)  -a_{0}(t)u\left(  x,t\right) \\
&  =\partial_{t}u\left(  x,t\right)  -\sum_{\alpha\in\mathcal{B}}a_{\alpha
}(t)\partial_{x}^{\alpha}u\left(  x,t\right)  \,,
\end{align*}
where $\mathcal{B}=\mathcal{A}\cup\left\{  \alpha_{0}\right\}  =\mathcal{A}%
\cup\left\{  \left(  0,\cdots,0\right)  \right\}  $. We let $\left\{
q_{\beta}\right\}  $ denote the polynomials (\ref{poly1}) for $\mathcal{L}%
_{0}$, and $\left\{  p_{\beta}\right\}  $ the corresponding polynomials for
$\mathcal{L}$. The analogue of (\ref{new1}) for $q_{\beta}$ is%
\begin{equation}
\frac{q_{\beta}\left(  x,t\right)  }{\beta!}=\sum_{\gamma+\overline{\alpha
}\cdot\sigma=\beta}\frac{x^{\gamma}b(t)^{\sigma}}{\gamma!\sigma!}\,,
\label{new17}%
\end{equation}
where here
\[
\overline{\alpha}\cdot\sigma=\sum_{\alpha\in\mathcal{B}}\alpha\sigma_{\alpha
}=\sum_{\alpha\in\mathcal{A}}\alpha\sigma_{\alpha}+\alpha_{0}\sigma_{0}%
=\sum_{\alpha\in\mathcal{A}}\alpha\sigma_{\alpha}\,.
\]
Thus the condition $\gamma+\overline{\alpha}\cdot\sigma=\beta$ places no
restriction on the component $\sigma_{0}$ of $\sigma$ corresponding to
$\alpha_{0}$; that is, for fixed acceptable choices of the other components
$\left\{  \sigma_{\alpha}\right\}  $ of $\sigma$, we may allow $\sigma_{0}$ to
range over all nonnegative integers. If we write%
\[
\sigma=\left(  \xi,\sigma_{0}\right)  =\left(  \xi_{1},\cdots,\xi_{K}%
,\sigma_{0}\right)  \,,\;\;\;b_{0}(t)=\int_{0}^{t}a_{0}(s)\,ds\,,
\]
then (\ref{new17}) may be written as%
\begin{equation}
\frac{q_{\beta}\left(  x,t\right)  }{\beta!}=\sum_{\gamma+\overline{\alpha
}\cdot\xi=\beta}\frac{x^{\gamma}b(t)^{\xi}}{\gamma!\xi!}\sum_{\sigma_{0}%
=0}^{\infty}\frac{b_{0}(t)^{\sigma_{0}}}{\left(  \sigma_{0}\right)  !}\,,
\label{new18}%
\end{equation}
where here we interpret%
\[
\overline{\alpha}\cdot\xi=\sum_{\alpha\in\mathcal{A}}\alpha\xi_{\alpha
}\,,\;\;\;\;b(t)^{\xi}=\prod_{\alpha\in\mathcal{A}}b_{\alpha}%
(t)^{\xi_{\alpha}}\,.
\]
Thus equation (\ref{new18}) confirms the formula%
\[
q_{\beta}(x,t)=p_{\beta}\left(  x,t\right)  \;e^{b_{0}(t)}\,.
\]
But $b_{0}(t)\leq\Lambda_{0}\left|  t\right|  $ where $\left|  a_{0}%
(t)\right|  \leq\Lambda_{0}$, giving%
\[
\left|  q_{\beta}\left(  x,t\right)  \right|  \leq e^{\Lambda_{0}\left|
t\right|  }\left|  p_{\beta}\left(  x,t\right)  \right|  \,.
\]
As $p_{\beta}$ has the bound (\ref{gen1}), the identical bound will hold for
$q_{\beta}$ because the multiplicative factor $e^{\Lambda_{0}\left|  t\right|
}$ only adds another term to the sum in the expression%
\[
\exp\left[  \sum_{\alpha\in\mathcal{A}\text{\ ,\ }\left|  \alpha\right|
<m}\left(  \frac{M+1}{\delta}\right)  ^{\left|  \alpha\right|  /\left(
m-\left|  \alpha\right|  \right)  }\left|  \Lambda_{\alpha}t\right|
^{m/\left(  m-\left|  \alpha\right|  \right)  }\right]  \,,
\]
so that $\alpha\in\mathcal{A}$ changes to $\alpha\in\mathcal{B}$. The number
$M$ of multi-indices $\alpha$ satisfying $0<\left|  \alpha\right|  <m$ is the
same for $\mathcal{A}$ and $\mathcal{B}$. These remarks complete the proof of
the theorem.
\end{proof}

When the operator (\ref{genop}) has no lower order terms, assuming the form%
\begin{equation}
\mathcal{L}u\left(  x,t\right)  =\partial_{t}u\left(  x,t\right)
-\sum_{\left|  \alpha\right|  =m}a_{\alpha}(t)\partial_{x}^{\alpha}u\left(
x,t\right)  \,, \label{homop}%
\end{equation}
the bound on the polynomials $\left\{  p_{\beta}\right\}  $ is simplified; we
have $M=0$, while sums over lower order multi-indices are $0$. Then in Theorem
\ref{genthm1} the bound (\ref{gen1}) reduces to%
\begin{equation}
\frac{\left|  p_{\beta}(x,t)\right|  }{\beta!}\leq\frac{\left(  \Lambda\left|
t\right|  +\delta\right)  ^{\left|  \beta\right|  /m}}{\left(  \left|
\beta\right|  /m\right)  !}\exp\left[  \frac{\left(  n\left|  x\right|
\right)  ^{m/(m-1)}}{\delta^{1/\left(  m-1\right)  }}\right]  \,.
\label{hom1}%
\end{equation}

\section{Series Expansions}

We consider polynomial series of the form%
\begin{equation}
u(x,t)=\sum_{\beta}\frac{c_{\beta}}{\beta!}p_{\beta}\left(  x,t\right)
\,, \label{series}%
\end{equation}
where again the polynomials $\left\{  p_{\beta}\right\}  $ are those
associated with the operator (\ref{genop}). Borrowing techniques from
\cite{KEM,RW,WID1,WID2,WID3}, we prescribe conditions on the coefficients
$\left\{  c_{\beta}\right\}  $ that guarantee convergence of the series in a
strip%
\begin{equation}
S_{s}=\left\{  \left(  x,t\right)  :x\in\mathbb{R}^{n},\;t\in\mathbb{I}%
,\;\left|  t\right|  <s\right\}  \;\;\;\text{.} \label{strip}%
\end{equation}
Theorem \ref{conv} below is our main tool for solving the Cauchy problem for
the operator $\mathcal{L}$ of (\ref{genop}), as described later in Theorem
\ref{cauchprob}. The analogous Cauchy problem was solved for the heat equation
by Rosenbloom and Widder, and subsequently by Kemnitz and Haimo and Markett
for other one dimensional equations, as mentioned earlier.

Given a real valued function $f$ defined on multi-indices $\beta$ in
$\mathbb{R}^{n}$, we define%
\[
\lim\sup_{\beta\rightarrow\infty}\;f\left(  \beta\right)  :=\lim
_{k\rightarrow\infty}\;\sup_{\left|  \beta\right|  \geq k}\;f\left(
\beta\right)  \,.
\]

\begin{theorem}
\label{conv}Under the hypotheses of Theorem \ref{genthm1}, assume that%
\begin{equation}
\lim\sup_{\beta\rightarrow\infty}\frac{\left|  c_{\beta}\right|  ^{m/\left|
\beta\right|  }me\Lambda}{\left|  \beta\right|  }=\frac{1}{s}\,,
\label{con1}%
\end{equation}
where $0<s\leq\infty$.

(a)\ \ Then the series (\ref{series}) converges absolutely and uniformly on
compact subsets of the strip $S_{s}$ to a function $u$.

(b)\ \ Both $u$ and $\partial u/\partial t$ are continuous in this strip, as
are their space derivatives $\partial_{x}^{\nu}u$ and $\partial_{x}^{\nu}%
u_{t}$ of all orders. All these derivatives can be obtained from
(\ref{series}) by termwise differentiation, with convergence of the
differentiated series likewise absolute and uniform on compact subsets of
$S_{s}$. Finally, $u$ solves in $S_{s}$ the equation $\mathcal{L}u=0$, where
$\mathcal{L}$ is the operator (\ref{genop}), while for each $\beta$,
\begin{equation}
c_{\beta}=\partial_{x}^{\beta}u(0,0)\,. \label{coefform}%
\end{equation}
\end{theorem}

\begin{proof}
Given $X,T$ with $0<X<\infty$, $0<T<s$, consider the region $R(X,T)$
containing all points $(x,t)$ such that
\[
\left(  x,t\right)  \in\mathbb{R}^{n}\times\mathbb{I}\,,\;\;\;\;\left|
x\right|  \leq X\,,\;\;\;\;\left|  t\right|  \leq T\,.
\]
Given $\delta>0$, inequality (\ref{gen1}) shows that, under the hypotheses of
Theorem \ref{genthm1}, there is a constant $C=C\left(  X,T,\delta\right)  $,
independent of $\beta$, such that for all $\left(  x,t\right)  \in R\left(
X,T\right)  $,%
\begin{equation}
\frac{\left|  p_{\beta}\left(  x,t\right)  \right|  }{\beta!}\leq
C\frac{\left(  \Lambda T+\delta\right)  ^{\left|  \beta\right|  /m}}{\left(
\left|  \beta\right|  /m\right)  !}\,. \label{st1}%
\end{equation}
By (\ref{g1}), in (\ref{st1}) we have%
\begin{equation}
\frac{1}{\left(  \left|  \beta\right|  /m\right)  !}\leq\frac{e^{\left|
\beta\right|  /m}}{\left(  \left|  \beta\right|  /m\right)  ^{\left|
\beta\right|  /m}}=\left(  \frac{em}{\left|  \beta\right|  }\right)  ^{\left|
\beta\right|  /m}\,. \label{st3}%
\end{equation}
Let $T_{1}$ satisfy $T<T_{1}<s$. Under assumption (\ref{con1}) there exists
$k>0$ such that $\left|  \beta\right|  \geq k$ implies%
\begin{equation}
\frac{\left|  c_{\beta}\right|  ^{m/\left|  \beta\right|  }me\Lambda}{\left|
\beta\right|  }\leq\frac{1}{T_{1}}\,. \label{st5}%
\end{equation}

We combine (\ref{st1}), (\ref{st3}), and (\ref{st5}) to conclude that, for
$\left|  \beta\right|  \geq k$ and under the hypotheses of Theorem
\ref{genthm1},
\begin{equation}
\left|  c_{\beta}\right|  \frac{\left|  p_{\beta}\left(  x,t\right)  \right|
}{\beta!}\leq C\left(  \frac{\Lambda T+\delta}{\Lambda T_{1}}\right)
^{\left|  \beta\right|  /m}\,. \label{st7}%
\end{equation}
As $\Lambda$ is positive and $T_{1}>T$, in (\ref{st7}) we can choose $\delta$
sufficiently small that the quanitity in parentheses on the right is less than
some positive constant $r$, with $r<1$; then we obtain%
\[
\sum_{\left|  \beta\right|  \geq k}\left|  \frac{c_{\beta}}{\beta!}p_{\beta
}\left(  x,t\right)  \right|  \leq C\sum_{\left|  \beta\right|  \geq
k}r^{\left|  \beta\right|  /m}<\infty\,.
\]
Thus (\ref{series}) converges absolutely and uniformly in the region $R\left(
X,T\right)  $ -- and likewise on compact subsets of $S_{s}$ because $X$ can be
arbitrarily large and $T$ arbitrarily close to $s$.

Keeping in mind the identity (\ref{oppol}), we consider a formal space
derivative of the series (\ref{series}),
\begin{align}
\partial_{x}^{\nu}u(x,t)  &  =\partial_{x}^{\nu}\sum_{\beta}\frac{c_{\beta}%
}{\beta!}p_{\beta}\left(  x,t\right)  =\sum_{\beta}\frac{c_{\beta}}{\beta
!}\partial_{x}^{\nu}p_{\beta}\left(  x,t\right) \label{dese}\\
&  =\sum_{\beta\geq\nu}\frac{c_{\beta}}{\left(  \beta-\nu\right)  !}%
p_{\beta-\nu}\left(  x,t\right)  =\sum_{\beta}\frac{c_{\beta+\nu}}{\beta
!}p_{\beta}\left(  x,t\right)  \,.\nonumber
\end{align}
Under assumption (\ref{con1}), for the coefficients in the derived series we
have%
\begin{align*}
&  \lim\sup_{\beta\rightarrow\infty}\frac{\left|  c_{\beta+\nu}\right|
^{m/\left|  \beta\right|  }me\Lambda}{\left|  \beta\right|  }\\
&  =\lim\sup_{\beta\rightarrow\infty}\left[  \frac{\left|  c_{\beta+\nu
}\right|  ^{m/\left|  \beta+\nu\right|  }me\Lambda}{\left|  \beta+\nu\right|
}\right]  ^{\left|  \beta+\nu\right|  /\left|  \beta\right|  }\frac{\left|
\beta+\nu\right|  }{\left|  \beta\right|  }\frac{\left|  \beta+\nu\right|
^{\left|  \nu\right|  /\left|  \beta\right|  }}{\left(  me\Lambda\right)
^{\left|  \nu\right|  /\left|  \beta\right|  }}\\
&  =\left(  \frac{1}{s}\right)  ^{1}\cdot1\cdot\frac{1}{1}=\frac{1}{s}\,.
\end{align*}
We may now apply the result of (a) to deduce that each derived series
(\ref{dese}) converges absolutely and uniformly on compact subsets of $S_{s}$.
This conclusion legitimizes our formal termwise differentiations of the
series, while also confirming the continuity of the sums.

Next, in view of (\ref{oppoly}), we compute the formal derivative of
(\ref{series})\ with respect to $t$ as
\begin{equation}
\partial_{t}\left(  \sum_{\beta}\frac{c_{\beta}}{\beta!}p_{\beta}(x,t)\right)
=\sum_{\beta}\sum_{\alpha\leq\beta}c_{\beta}a_{\alpha}(t)\frac{p_{\beta
-\alpha}(x,t)}{\left(  \beta-\alpha\right)  !}\,, \label{st9}%
\end{equation}
where%
\[
\sum_{\beta}\sum_{\alpha\leq\beta}\left|  c_{\beta}a_{\alpha}(t)\frac{p_{\beta
-\alpha}(x,t)}{\left(  \beta-\alpha\right)  !}\right|  =\sum_{\alpha}\left|
a_{\alpha}(t)\right|  \sum_{\beta}\left|  c_{\beta+\alpha}\right|
\frac{\left|  p_{\beta}\left(  x,t\right)  \right|  }{\beta!}\,.
\]
As above, each of the series
\[
\sum_{\beta}\left|  c_{\beta+\alpha}\right|  \frac{\left|  p_{\beta
}(x,t)\right|  }{\beta!}%
\]
converges uniformly and absolutely on compact subsets of $S_{s}$; as the
coefficients $\left\{  a_{\alpha}(t)\right\}  $ are bounded, the same is true
for the series (\ref{st9}). Thus termwise differentiation is justified in
(\ref{st9}), and the sum of the series is continuous in $S_{s}$.

Applying again (\ref{oppol}) and (\ref{oppoly}), we compute also formal mixed
derivatives of (\ref{series}),%
\[
\partial_{t}\partial_{x}^{\nu}\sum_{\beta}\frac{c_{\beta}}{\beta!}p_{\beta
}\left(  x,t\right)  =\partial_{x}^{\nu}\partial_{t}\sum_{\beta}%
\frac{c_{\beta}}{\beta!}p_{\beta}\left(  x,t\right)  =\sum_{\beta}\sum
_{\alpha\leq\beta}c_{\beta+\nu}a_{\alpha}\left(  t\right)  \frac{p_{\beta
-\alpha}\left(  x,t\right)  }{\left(  \beta-\alpha\right)  !}\,,
\]
where%
\[
\sum_{\beta}\sum_{\alpha\leq\beta}\left|  c_{\beta+\nu}a_{\alpha}\left(
t\right)  \frac{p_{\beta-\alpha}\left(  x,t\right)  }{\left(  \beta
-\alpha\right)  !}\right|  =\sum_{\alpha}\left|  a_{\alpha}(t)\right|
\sum_{\beta}\left|  c_{\beta+\alpha+\nu}\right|  \frac{\left|  p_{\beta
}\left(  x,t\right)  \right|  }{\beta!}\,.
\]
By arguments above, these series likewise converge absolutely and uniformly on
compact subsets of $S_{s}$, and termwise differentiation and continuity of the
sums are justified.

The formula $\mathcal{L}u=0$ follows by termwise differentiation and the fact
that $\mathcal{L}p_{\beta}=0$ for each $\beta$. Finally, setting $\left(
x,t\right)  =\left(  0,0\right)  $ in (\ref{dese}) and using $p_{\beta}\left(
x,0\right)  =x^{\beta}$ gives $\partial_{x}^{\nu}u\left(  0,0\right)  =c_{\nu
}$, and thereby (\ref{coefform}).
\end{proof}

It is noteworthy in formula (\ref{con1}) that bounds on lower order terms do
not appear -- only highest order bounds affect the estimate on the width of
the strip of convergence.

For the special case of the Kemnitz operator in one space dimension,%
\[
\partial_{t}u\left(  x,t\right)  -\partial_{x}^{m}u\left(  x,t\right)
\,,
\]
we have $\Lambda=1$ and (\ref{con1}) becomes the estimate of Kemnitz
\cite{KEM}.

\section{Cauchy Problems}

Again we let $\mathcal{L}$ denote the operator (\ref{genop}). We consider
Cauchy problems%
\begin{equation}
\left\{
\begin{array}
[c]{c}%
\mathcal{L}u(x,t)=0\,,\\
u\left(  x,0\right)  =f\left(  x\right)  \,.
\end{array}
\right.  \label{cp}%
\end{equation}
We shall assume $f$ is a real valued function on $\mathbb{R}^{n}$, with a
power series expansion in $\mathbb{R}^{n}$ which we write in the two ways%
\begin{equation}
f\left(  x\right)  =\sum_{\beta}a_{\beta}x^{\beta}=\sum_{\beta}\frac{c_{\beta
}}{\beta!}x^{\beta}\,. \label{exp}%
\end{equation}

We introduce some terminology regarding growth conditions on the function $f$.
We state the conditions for complex valued analytic functions defined in
$\mathbb{C}^{n}$, as they are more natural in that setting. Whereas for entire
functions of a single complex variable there is general agreement on
definitions of order and type (see \cite{BOA}, for example), for functions of
several complex variables there is no standard terminology, as there are
several possible definitions depending on one's choice of norms in
$\mathbb{C}^{n}$ and on other considerations. (See \cite{GL}, Chapter 1, for a
discussion of this topic.) Although it appears that all definitions give the
same order of an entire function, the type varies with the definition. The
definitions below are most suitable for our purposes, as they allow the
natural extension to higher space dimensions of one-dimensional results of
Rosenbloom and Widder, and later of Kemnitz, regarding the Cauchy problem
(\ref{cp}). We explain in Theorem \ref{grow} how the prescribed conditions on
the coefficients of the power series expansion of an entire function are
equivalent to growth limits at infinity on the function.

\begin{definition} \label{defgrow} \rm
Let $f=f\left(  z\right)  =f\left(  z_{1},z_{2},\cdots
,z_{n}\right)  $ be an entire function $f$ in $\mathbb{C}^{n}$, with the power
series expansion (with complex coefficients)%
\begin{equation}
f\left(  z\right)  =\sum_{\beta}a_{\beta}z^{\beta}=\sum_{\beta}\frac{c_{\beta
}}{\beta!}z^{\beta}\,. \label{expb}%
\end{equation}
Given $\rho,\tau$ with $0<\rho,\tau<\infty$, we say that $f$ has growth
$\left\{  \rho,\tau\right\}  $ provided that%
\begin{equation}
\lim\sup_{\beta\rightarrow\infty}\;\left|  \beta\right|  ^{1-\rho}\left(
\beta^{\beta/\left|  \beta\right|  }\right)  ^{\rho}\left|  a_{\beta}\right|
^{\rho/\left|  \beta\right|  }\leq e\tau\rho\,. \label{gt1}%
\end{equation}
\end{definition}

\begin{theorem}
\label{grow}Assume $0<\rho,\tau<\infty$, and that $a_{\beta}=c_{\beta}/\beta!$
for each $\beta$. Then condition (\ref{gt1}) is equivalent to%
\begin{equation}
\lim\sup_{\beta\rightarrow\infty}\;\left(  \frac{e}{\left|  \beta\right|
}\right)  ^{\rho-1}\left|  c_{\beta}\right|  ^{\rho/\left|  \beta\right|
}\leq\tau\rho\,, \label{gt3}%
\end{equation}
and under these conditions on the coefficients the series (\ref{expb})
converges to an entire function $f$ in $\mathbb{C}^{n}$ satisfying the growth
condition%
\begin{equation}
\left|  f\left(  z\right)  \right|  =\mathcal{O}\left(  \exp\tau^{\prime
}\left[  \sum_{i=1}^{n}\left|  z_{i}\right|  \right]  ^{\rho}\right)
\;\;\;\text{as }z\rightarrow\infty\text{,\ for all }\tau^{\prime}>\tau\,.
\label{gt4}%
\end{equation}
Conversely, if $f$ is an entire function in $\mathbb{C}^{n}$ with growth
condition (\ref{gt4}) and the expansion (\ref{expb}), then (\ref{gt1}) and
(\ref{gt3}) must hold.
\end{theorem}

Before proving Theorem \ref{grow} we demonstrate how the coefficient
conditions of that theorem pertain to the Cauchy problem (\ref{cp}). A real
valued function $f$ with an expansion (\ref{exp}) in $\mathbb{R}^{n}$ can be
viewed as the restriction to $\mathbb{R}^{n}$ of an analytic function in
$\mathbb{C}^{n}$ with the expansion (\ref{expb}). Accordingly, we may apply
Definition \ref{defgrow} just as well to real valued functions with expansions
(\ref{exp}) valid in $\mathbb{R}^{n}$.

\begin{theorem}
\label{cauchprob}Under the hypotheses of Theorem \ref{genthm1} assume that the
function (\ref{exp}) has growth $\left\{  m/\left(  m-1\right)  ,\tau\right\}
$ for some $\tau$, $0<\tau<\infty$. Then the series%
\begin{equation}
u\left(  x,t\right)  =\sum_{\beta}\frac{c_{\beta}}{\beta!}p_{\beta}\left(
x,t\right)  \label{sol}%
\end{equation}
solves the Cauchy problem (\ref{cp}) in the strip%
\[
S_{s}=\left\{  \left(  x,t\right)  :x\in\mathbb{R}^{n}\,,\;t\in\mathbb{I}%
\,,\;\left|  t\right|  <s\right\}  \,,
\]
where%
\begin{equation}
s=\frac{1}{m\Lambda}\left(  \frac{m-1}{m\tau}\right)  ^{m-1}\,.
\label{str1}%
\end{equation}
Moreover, the series (\ref{sol}) and its sum $u\left(  x,t\right)  $ have the
convergence and regularity properties described in Theorem \ref{conv}.
\end{theorem}

\begin{proof}
Under the hypotheses of Theorem \ref{genthm1}, the growth condition
(\ref{gt3}) on $f$, with $\rho=\left(  m-1\right)  /m$, implies%
\[
\lim\sup_{\beta\rightarrow\infty}\;\frac{\left|  c_{\beta}\right|  ^{m/\left|
\beta\right|  }me\Lambda}{\left|  \beta\right|  }\leq\frac{1}{s}\,,
\]
where $s$ is prescribed by (\ref{str1}).\ Thus Theorem \ref{conv} applies,
confirming the stated convergence properties of the series (\ref{sol}) and the
regularity properties of its sum $u\left(  x,t\right)  $, and affirming as
well the equation $\mathcal{L}u=0$ in $S_{s}$. The initial value of $u$ is%
\[
u\left(  x,0\right)  =\sum_{\beta}\frac{c_{\beta}}{\beta!}p_{\beta}\left(
x,0\right)  =\sum_{\beta}\frac{c_{\beta}}{\beta!}x^{\beta}=f(x)\,.
\]
\end{proof}

The proof of Theorem \ref{grow} requires a technical lemma.

\begin{lemma}
\label{tech}If $0<r,\rho<\infty$, then%
\begin{equation}
\sum_{k=0}^{\infty}\frac{r^{k}}{k^{k/\rho}}\leq\left(  2^{\rho}r^{\rho
}+2\right)  \exp\left(  \frac{r^{\rho}}{e\rho}\right)  \,. \label{gr1}%
\end{equation}
\end{lemma}

\begin{proof}
The function $\psi(k)=r^{k}k^{-k/\rho}$ attains a maximum on $[0,\infty)$ at
$k=r^{\rho}/e$; thus%
\begin{equation}
r^{k}k^{-k/\rho}\leq\psi\left(  r^{\rho}/e\right)  =\exp\left(  \frac{r^{\rho
}}{e\rho}\right)  \,. \label{gr2}%
\end{equation}
The sum on the left of (\ref{gr1}) may be written as $S_{1}+S_{2}$, where
\[
S_{1}=\sum_{k<2^{\rho}r^{\rho}}\frac{r^{k}}{k^{k/\rho}}%
\,,\;\;\;\;\;S_{2}=\sum_{k\geq2^{\rho}r^{\rho}}\frac{r^{k}}{k^{k/\rho
}}\,.
\]
In $S_{1}$ we bound each term by (\ref{gr2}) to obtain%
\[
S_{1}\leq\sum_{k<2^{\rho}r^{\rho}}\exp\left(  \frac{r^{\rho}}{e\rho}\right)
\leq\left(  2^{\rho}r^{\rho}+1\right)  \exp\left(  \frac{r^{\rho}}{e\rho
}\right)  \,.
\]
In $S_{2}$ we have $r\leq k^{1/\rho}/2$, and thus
\[
S_{2}\leq\sum_{k\geq2^{\rho}r^{\rho}}\frac{\left(  k^{1/\rho}/2\right)  ^{k}%
}{k^{k/\rho}}\leq\sum_{k=1}^{\infty}2^{-k}=1\leq\exp\left(  \frac{r^{\rho}%
}{e\rho}\right)  \,.
\]
Combining the estimates for $S_{1}$ and $S_{2}$ gives (\ref{gr1}).
\end{proof}


Now we prove Theorem \ref{grow}. We use techniques adapted from similar
arguments in \cite{BOA,GL,WID1}.

First, from (\ref{g1}) it follows that for any multi-index $\beta$ in
$\mathbb{R}^{n}$,
\[
\beta^{\beta}e^{-\left|  \beta\right|  }\leq\beta!\leq\beta^{\beta}e^{-\left|
\beta\right|  }e^{n}\prod_{i=1}^{n}\sqrt{\beta_{i}+1}\,.
\]
From this inequality and the observation%
\[
\lim_{\left|  \beta\right|  \rightarrow\infty}\;\left(  e^{n}\prod_{i=1}%
^{n}\sqrt{\beta_{i}+1}\right)  ^{\rho/\left|  \beta\right|  }=1\,,
\]
it is readily checked that (\ref{gt1}) is equivalent to (\ref{gt3}).

Now we assume condition (\ref{gt3}) holds on the coefficients of (\ref{expb}),
and verify the series converges to an entire function $f$ in $\mathbb{C}^{n}$
satisfying (\ref{gt4}). Let $\delta$ be fixed, $\delta>0$. Then (\ref{gt3})
guarantees that for $\left|  \beta\right|  $ sufficiently large,%
\[
\left|  c_{\beta}\right|  \leq\left[  \left(  \tau+\delta\right)  \rho\right]
^{\left|  \beta\right|  /\rho}\left(  \left|  \beta\right|  /e\right)
^{\left|  \beta\right|  -\left|  \beta\right|  /\rho}\,.
\]
But the right hand side of this inequality is bounded below on any finite
collection of $\beta^{\prime}s$ (as we interpret $0^{0}=1$), so for some
constant $M$ we have for all $\beta$ that%
\[
\left|  c_{\beta}\right|  \leq M\left[  \left(  \tau+\delta\right)
\rho\right]  ^{\left|  \beta\right|  /\rho}\left(  \left|  \beta\right|
/e\right)  ^{\left|  \beta\right|  -\left|  \beta\right|  /\rho}\,.
\]
Then for the series (\ref{expb}) we have the bound%
\begin{align*}
\left|  f\left(  z\right)  \right|   &  \leq M\sum_{\beta}\left[  \left(
\tau+\delta\right)  \rho\right]  ^{\left|  \beta\right|  /\rho}\left(  \left|
\beta\right|  /e\right)  ^{\left|  \beta\right|  -\left|  \beta\right|  /\rho
}\frac{\left|  z^{\beta}\right|  }{\beta!}\\
&  =M\sum_{k=0}^{\infty}\frac{\left[  e\rho\left(  \tau+\delta\right)
\right]  ^{k/\rho}k^{k-k/\rho}}{e^{k}}\sum_{\left|  \beta\right|
=k}\frac{\left|  z^{\beta}\right|  }{\beta!}\,.
\end{align*}
Writing $\left\|  z\right\|  _{1}:=\sum_{i=1}^{n}\left|  z_{i}\right|  $, we
have%
\[
\left(  \left\|  z\right\|  _{1}\right)  ^{k}=\sum_{\left|  \beta\right|
=k}\frac{k!}{\beta!}\prod_{i=1}^{n}\left|  z_{i}\right|  ^{\beta_{i}}%
=k!\sum_{\left|  \beta\right|  =k}\frac{\left|  z^{\beta}\right|  }{\beta
!}\,,
\]
and thereby%
\[
\left|  f\left(  z\right)  \right|  \leq M\sum_{k=0}^{\infty}\frac{\left[
e\rho\left(  \tau+\delta\right)  \right]  ^{k/\rho}}{k^{k/\rho}}%
\frac{k^{k}\left(  \left\|  z\right\|  _{1}\right)  ^{k}}{e^{k}k!}\,.
\]
By (\ref{g1}), $e^{k}k!\geq k^{k}$, and consequently%
\[
\left|  f\left(  z\right)  \right|  \leq M\sum_{k=0}^{\infty}\frac{\left\{
\left[  e\rho\left(  \tau+\delta\right)  \right]  ^{1/\rho}\left\|  z\right\|
_{1}\right\}  ^{k}}{k^{k/\rho}}\,.
\]
Finally, by Lemma \ref{tech} the series on the right converges uniformly on
compact subsets of $\mathbb{C}^{n}$, and%
\begin{align*}
\left|  f\left(  z\right)  \right|   &  \leq M\left[  2^{\rho}e\rho\left(
\tau+\delta\right)  \left(  \left\|  z\right\|  _{1}\right)  ^{\rho}+2\right]
\exp\left[  \left(  \tau+\delta\right)  \left(  \left\|  z\right\|
_{1}\right)  ^{\rho}\right] \\
&  =\mathcal{O}\left(  \exp\left[  \left(  \tau+2\delta\right)  \left(
\left\|  z\right\|  _{1}\right)  ^{\rho}\right]  \right)  \,.
\end{align*}
Since $\delta$ can be arbitrarily small, (\ref{gt4}) follows. Clearly $f$, the
sum in $\mathbb{C}^{n}$ of a power series, is entire.

Next suppose $f$ is entire in $\mathbb{C}^{n}$ with the expansion
(\ref{expb}), and that (\ref{gt4}) holds. We verify (\ref{gt1}). Given
$\tau^{\prime}>\tau$, (\ref{gt4}) implies there is a positive constant $M$
such that%
\[
\left|  f\left(  z\right)  \right|  \leq M\exp\left[  \tau^{\prime}\left(
\left\|  z\right\|  _{1}\right)  ^{\rho}\right]  \,.
\]
By the generalized Cauchy inequality for analytic functions in $\mathbb{C}%
^{n}$ (see \cite{HOR}, Theorem 2.2.7), for any multi-index $\beta$ and
positive vector $r=\left(  r_{1},\cdots,r_{n}\right)  $,%
\begin{align}
\left|  c_{\beta}\right|   &  =\left|  \partial^{\beta}f\left(  0\right)
\right|  \leq\frac{\beta!}{r^{\beta}}\sup\left\{  \left|  f\left(  z\right)
\right|  :\left|  z_{i}\right|  \leq r_{i}\,,\;1\leq i\leq n\right\}
\label{cauchy}\\
&  \leq\frac{M\beta!}{r^{\beta}}\exp\left[  \tau^{\prime}\left(  \sum
_{i=1}^{n}r_{i}\right)  ^{\rho}\right]  \,.\nonumber
\end{align}
The gradient with respect to $r$ of the expression on the right vanishes when%
\begin{equation}
r_{i}=\frac{\beta_{i}}{\rho^{1/\rho}\left(  \tau^{\prime}\right)  ^{1/\rho
}\left|  \beta\right|  ^{\left(  \rho-1\right)  /\rho}}\,,\;1\leq i\leq
n\;\;; \label{setr}%
\end{equation}
we choose these values of $r_{i}$ and obtain%
\begin{equation}
\left|  a_{\beta}\right|  =\left|  c_{\beta}\right|  /\beta!\leq\frac{M\left(
e\tau^{\prime}\rho\right)  ^{\left|  \beta\right|  /\rho}\left|  \beta\right|
^{\left|  \beta\right|  -\left|  \beta\right|  /\rho}}{\beta^{\beta}}\,.
\label{setrb}%
\end{equation}
(In the event that $\beta_{i}=0$ for some $i$, we replace $\beta_{i}$ with
$\beta_{i}+\varepsilon_{i}$ in the numerator on the right side of
(\ref{setr}), as we require $r_{i}>0$; but then (\ref{setrb}) follows by
letting $\varepsilon_{i}\rightarrow0^{+}$.) From (\ref{setrb}) it follows that%
\[
\lim\sup_{\beta\rightarrow\infty}\;\left|  \beta\right|  ^{1-\rho}\left(
\beta^{\beta/\left|  \beta\right|  }\right)  ^{\rho}\left|  a_{\beta}\right|
^{\rho/\left|  \beta\right|  }\leq\lim\sup_{\beta\rightarrow\infty}%
\;M^{\rho/\left|  \beta\right|  }e\tau^{\prime}\rho=e\tau^{\prime}\rho\,.
\]
As this inequality holds for all $\tau^{\prime}>\tau,$ we have verified
(\ref{gt1}).

\begin{thebibliography}{9}                                                                                                %

\bibitem {AP}P. Appell, \emph{Sur \v{l}\'{e}quation} $\partial^{2}z/\partial
x^{2}-\partial z/\partial y=0$ \emph{et la th\'{e}orie de la chaleur}, J.
Math. Pures Appl. 8 (1892), 187-216.

\bibitem {ART}E. Artin, The \emph{Gamma Function}, Holt, Rinehart, and
Winston, 1964.

\bibitem {BOA}R. P. Boas, $\emph{Entire\;Functions}$, Academic Press, New
York, 1954.

\bibitem {BRA}L. R. Bragg, \emph{The radial heat polynomials and related
functions}, Trans. Amer. Math. Soc. 119 (1965), 270-290.

\bibitem {BD1}L. R. Bragg and J. W. Dettman, \emph{Expansions of solutions of
certain hyperbolic and elliptic problems in terms of Jacobi polynomials}, Duke
Math. J. 36 (1969), 129-144.

\bibitem {BD2}L. R. Bragg and J. W. Dettman, \emph{Analogous function theories
for the heat, wave, and Laplace equations}, Rocky Mountain J. Math. 13, No. 2
(1983), 191-214.

\bibitem {CH}F. M. Cholewinski and D. T. Haimo, \emph{Expansions in terms of
Laguerre heat polynomials and of their temperature transforms}, J. Math. Anal.
24 (1971), 285-322.

\bibitem {COL}D. Colton, \emph{Analytic Theory of Partial Differential
Equations}, Pitman Publishing, Boston, 1980.

\bibitem {FIT}A. Fitouhi, \emph{Heat polynomials for a singular operator on
(0,}$\infty\emph{)}$, Constr. Approx. 5 (1989), 241-270.

\bibitem {GL}L. Gruman and P. Lelong, \emph{Entire Functions of Several
Complex Variables}, Springer-Verlag, 1986.

\bibitem {HAI0}D. T. Haimo, $\emph{L}^{2}$\emph{\ expansions in terms of
generalized heat polynomials and of their Appell transforms}, Pacific J. Math.
15 (1965), 865-875.

\bibitem {HAI1}D. T. Haimo, \emph{Expansions in terms of generalized heat
polynomials and of their Appell transforms}, J. Math. Mechan. 15 (1966), 735-758.

\bibitem {HAI2}D. T. Haimo, \emph{Generalized temperature functions}, Duke
Math. J. 33 (1966), 305-322.

\bibitem {HAI3}D. T. Haimo, \emph{Series expansions of generalized temperature
functions in N dimensions}, Canad. J. Math. 18 (1966), 794-802.

\bibitem {HAI4}D. T. Haimo, \emph{Series representation of generalized
temperature functions}, SIAM J. Appl. Math. 15 (1967), 359-367.

\bibitem {HAI5}D. T. Haimo, \emph{Series expansions and integral
representations of generalized temperatures}, Illinois J. Math. 14 (1970), 621-629.

\bibitem {HAI6}D. T. Haimo, \emph{Homogeneous generalized temperatures}, SIAM
J. Math. Anal. 11 (1980), 473-487.

\bibitem {HM1}D. T. Haimo and C. Markett, \emph{A representation theory for
solutions of a higher order heat equation, I}, J. Math. Anal. Appl. 168
(1992), 89-107.

\bibitem {HM2}D. T. Haimo and C. Markett, \emph{A representation theory for
solutions of a higher order heat equation, II}, J. Math. Anal. Appl. 168
(1992), 289-305.

\bibitem {HM}G. N. Hile and C. P. Mawata, \emph{The behaviour of the heat
operator on weighted Sobolev spaces}, Trans. Amer. Math. Soc. 350, No. 4
(1998), 1407-1428.

\bibitem {HS}G. N. Hile and A. Stanoyevitch, \emph{Heat polynomial analogs for
higher order evolution equations}, Electronic Journal of Differential
Equations 2001 (2001), No. 28, 1-19.

\bibitem {HOR}L. H\"{o}rmander, \emph{An Introduction to Complex Analysis in
Several Variables}, D. Van Nostrand, Princeton, N. J., 1966.

\bibitem {KAR1}V. V. Karachik,\ \emph{Continuity of polynomial solutions with
respect to the coefficient of the highest order derivative}, Indian J. Pure
Appl. Math. 28(9) (1997), 1229-1234.

\bibitem {KAR2}V. V. Karachik, \emph{On one set of orthogonal harmonic
polynomials}, Proc. Amer. Math. Soc. 126, No. 12 (1998), 3513-3519.

\bibitem {KAR3}V. V. Karachik, \emph{Polynomial solutions to systems of
partial differential equations with constant coefficients}, Yokohama Math. J.
47 (2000), 121-142.

\bibitem {KEM}H. Kemnitz, \emph{Polynomial expansions for solutions of}
$D_{x}^{r}u(x,t)=D_{t}u(x,t)$, $t=2,3,4,\ldots$, SIAM J. Math. Anal. 13
(1982), 640-650.

\bibitem {LEB}N. N. Lebedev, \emph{Special Functions and their Applications},
Prentice-Hall, 1965.

\bibitem {LO}C. Y. Lo, \emph{Polynomial expansions of solutions of }%
$u_{xx}+\varepsilon^{2}u_{tt}=u_{t}$, Z. Reine Angew. Math. 253 (1972), 88-103.

\bibitem {PED1}P. Pedersen, \emph{Analytic solutions for a class of pde's with
constant coefficients}, J. Differential Integral Equations 3, No. 4 (1990), 721-732.

\bibitem {PED2}P. Pedersen, \emph{A basis for polynomial solutions to systems
of linear constant coefficient pde's}, Advances in Math. 117, No. 1 (1996), 157-163.

\bibitem {PED3}P. Pedersen, \emph{A function theory for finding a basis for
all polynomial solutions to linear constant coefficient pde's of homogeneous
order}, Complex Variables 24 (1993), 79-87.

\bibitem {RW}P. C. Rosenbloom and D. V. Widder, \emph{Expansions in terms of
heat polynomials and associated functions}, Trans. Amer. Math. Soc. 92 (1959), 220-266.

\bibitem {SM}S. P. Smith, \emph{Polynomial solutions to constant coefficient
differential equations}, Trans. Amer. Math. Soc. 329, No. 2 (1992), 551-569.

\bibitem {WW}E. T. Whittaker and G. N. Watson, \emph{A Course of Modern
Analysis}, third edition, Cambridge University Press, 1920.

\bibitem {WID1}D. V. Widder, \emph{Series expansions of solutions of the heat
equation in n dimensions}, Ann. Mat. Pura Appl. 55 (1961), 389-410.

\bibitem {WID2}D. V. Widder, \emph{Expansions in series of homogeneous
temperature functions of the first and second kinds}, Duke Math. J. 36 (1969), 495-510.

\bibitem {WID3}D. V. Widder, \emph{The Heat Equation}, Academic Press, New
York, San Francisco, London, 1975.

\end{thebibliography}
\end{document}