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\title{Taylor Series Are Limits of Legendre Expansions}
\author{Paul E. Fishback\\
Department of Mathematics \\
Grand Valley State University \\
Allendale, Michigan 49401}
\date{December 8, 2004}
\maketitle
\newpage
\begin{center}
\Large{Taylor Series Are Limits of Legendre Expansions}
\end{center}

\paragraph*{Introduction.}
In \cite{Askey}, Askey and Haimo pose the following question:
\begin{quote}
``What are some basic similarities between power series and Fourier series beyond the fact that they are both 
infinite series?''
\end{quote}
A slight generalization of this question might consider comparing power series and other types of series expansions
involving orthogonal functions. Perhaps the simplest type of such a series expansion is a Legendre series. Such
series are constructed using the classical Legendre polynomials:
\begin{align}
P_0(x)&=1 \notag \\
P_1(x)&=x \notag \\
P_2(x)&=\frac{1}{2}(3x^2-1)\notag \\
P_3(x)&=\frac{1}{2}(5x^3-3x)\notag \\
   &\vdots \notag \\
P_n(x)&= \frac{2n-1}{n}P_{n-1}(x)-\frac{n-1}{n}P_{n-2}(x).\notag
\end{align}
Each Legendre polynomial $P_n$ is a solution of the second order Sturm-Liouville differential equation
	\[ \frac{d}{dx}\left[ (1-x^2)\frac{dy}{dx}\right] + n(n+1)y=0.\]
Taken together, the family of functions $\displaystyle \left\{ \sqrt{\frac{2n+1}{2}}P_n \right\}_{n=0}^{n=\infty}$ forms a complete
orthonormal basis for the Hilbert space $L^2\left([-1,1]\right)$.

\newpar
A classical result first published in 1862 by K. Neumann (\cite{Neumann}), but
also cited as a special case of Theorem 9.1.1 in \cite{Szego}, states that for a function
$f$ analytic in an open disk of radius $R>1$, the 
Legendre series of $f$, 
	\[
	\sum_{n=0}^{n=\infty} \left( \frac{2n+1}{2} \int_{t=-1}^{t=1} f(t)
	P_n(t)\,dt \right) P_n(z), \]
converges to $f(z)$ at all $z$ interior to the largest ellipse having foci
at $(\pm 1,0)$ in which $f$ is analytic. In addition, if $\rho$ denotes the 
sum of the semiaxes of this ellipse, then 	
\[ \frac{1}{\rho}= {\overline{\lim}}_{n \to \infty} \left|\frac{2n+1}{2} \int_{t=-1}^{t=1} f(t)
	P_n(t)\,dt   \right|^{\frac{1}{n}} .\]
This formula is analogous to the Cauchy-Hadamard Formula for the radius of convergence of a power series.
By considering the smallest possible ellipse above, which is inscribed 
in the circle $|z|=R$, one obtains the inequality
	%\[  \frac{1}{\sqrt{R^2+1}+R} \leq 
	\[ {\overline{\lim}}_{n \to \infty} \left|\frac{2n+1}{2} \int_{t=-1}^{t=1} f(t)
	P_n(t)\,dt   \right|^{\frac{1}{n}} \leq \frac{1}{R+\sqrt{R^2-1}}.\] 


\newpar
If instead, one only assumes that $f$ is analytic in a disk of radius $R>0$ about the origin, then for $h$ sufficiently small and positive,
$\frac{R}{h}>1$, and $f$
possesses a Legendre series relative to the interval $[-h,h]$ given by 
	\[
	\sum_{n=0}^{n=\infty} \left( \frac{2n+1}{2h} \int_{t=-h}^{t=h} f(t)
	P_n(t/h)\,dt \right) P_n(z/h). \]
(See \cite{Sansone}.) A simple change of variable, together with Neumann's result above, implies that this series
converges to $f(z)$ at all $z$ interior to an ellipse centered at the origin and having foci at 
$(\pm h,0)$. Moreover, 
\begin{equation}{\label{E:coeff_bound}} 
	{\overline{\lim}}_{n \to \infty} \left|\frac{2n+1}{2h} \int_{t=-h}^{t=h} f(t)
	P_n(t/h)\,dt   \right|^{\frac{1}{n}} \leq \frac{1}{\frac{R}{h}+\sqrt{\left(\frac{R}{h}\right)^2-1} }.
\end{equation}

\paragraph*{A Limit Theorem.}
In one sense then, a fundamental difference exists between the Taylor and Legendre series expansion, with one being valid in
a disk centered at the origin and the other in an ellipse having foci at $(\pm h,0)$. However, as $h$ approaches 
zero, the foci of this ellipse move toward the origin. Given this fact, one might wonder whether the corresponding limiting
value of the Legendre series is simply the Taylor series. The following theorem establishes that this is precisely the case. Surprisingly,
a proof of this somewhat intuitively obvious result, to the best of the author's knowledge, is not present in the literature.

\begin{theorem} For $f$ analytic in the open disk of radius $R>0$ at the origin and for $z$ interior to this disk,
\begin{equation}{\label{E:main_result}}
	 \lim_{h \to 0^+} \sum_{n=0}^{n=\infty} \left( \frac{2n+1}{2h} \int_{t=-h}^{t=h} f(t)
	P_n(t/h)\,dt \right) P_n(z/h) = 
	\sum_{n=0}^{n=\infty} \frac{f^{(n)}(0)}{n!}z^n. 
\end{equation}
In other words, the limiting value of the Legendre series expansion at $z$ is merely
the Taylor series expansion.  
%UNIFORM CONVERGENCE AND WEIERSTRASS M-TEST?
\end{theorem}
\begin{proof}
Define the function
$g(h)=\frac{|z|+h}{R+\sqrt{R^2-h^2}}$, which is continuous
on $[0,R]$ and satisfies 
$g(0)=\frac{|z|}{2R}<\frac{1}{2}$. By continuity, there exists $0<h_0 <R$ so that 
for all $0<h \leq h_0$, $g(h)\leq \frac{r}{2}$ for some fixed $r$ in $[0,1)$. 

\newpar
The proof will proceed in two primary steps. In the first step, we will assume $h < h_0$ and
establish that
\[ {\overline{\lim}}_{n \to \infty} \left| \left(\frac{2n+1}{2h} \int_{t=-h}^{t=h} f(t)
	P_n(t/h)\,dt \right) P_n(z/h)  \right|^{\frac{1}{n}}\leq r <1. \]
This inequality will justify passing the limit as $h \to 0^+$ inside the series above.

%\newpar
%FRACTIONAL PART and citation ?
%By the triangle inequality and the identity  
%	\[ P_n(z)=\frac{1}{2^n}\sum_{k=1}^{ \lfloor \frac{n}{2} \rfloor } (-1)^k \begin{pmatrix} n \\ k \end{pmatrix} 
%		\begin{pmatrix} 2n-2k \\ n \end{pmatrix} z^{n-2k},  \quad\mbox{(See \cite{Sansone})  } \]
%it follows that $ \left|P_n(z/h)\right| \leq \left|P_n(|z|/h)\right|= \left|P_n(|z|/h)\right|$. 
%Since equality is possible for $z$ purely imaginary, we must must now bound $|P_n|$ at $z/h$ by bounding this same quantity at $i\frac{|z|}{h}$.

\newpar
%CHECK BELOW FOR ACCURACY
The key ingredient in this process will be a result of Mauro Picone (\cite{Picone}) concerning the maximum
modulus of $P_n$ at a complex input $w$. If $\epsilon$ denotes the eccentricity
of the ellipse having center at the origin, foci at $(\pm 1,0)$, and major axis of length $2|w|$, then 
	\[ \left| P_n(w) \right| \leq \frac{(2n-1)!!}{n!}  \left(\frac{1}{\epsilon}+1\right)^n.\]
Here the double factorial $(2n-1)!!$ denotes $(2n-1)(2n-3)\cdots 3\cdot 1$.
Applying Picone's result and simplifying the resulting eccentricity to $|z|/h$, we arrive at 
 \[ \left|P_n(z/h)\right| \leq \frac{ (2n-1)!!}{n!}  \left(\frac{|z|}{h} +1\right) ^n.\]

%\newpage
This inequality, combined with (\ref{E:coeff_bound}), yields

\[ {\overline{\lim}}_{n \to \infty} \left| \left( \frac{2n+1}{2h} \int_{t=-h}^{t=h} f(t)
	P_n(t/h)\,dt \right) P_n(z/h)  \right|^{\frac{1}{n}}  \hspace{3in} \]
\begin{align} 
&\leq {\overline{\lim}}_{n \to \infty} \left( \frac{1}{ \frac{R}{h}+\sqrt{\left(\frac{R}{h}\right)^2-1} }         \right) \cdot
	\left( \frac{ (2n-1)!!}{n!}\right)^{\frac{1}{n}}  \left(\frac{|z|}{h} +1\right)  \notag \\
&={\overline{\lim}}_{n \to \infty}  \left( \frac{ (2n-1)!!}{n!}\right)^{\frac{1}{n}}  \cdot \frac{|z|+h}{R+\sqrt{R^2-h^2}} \notag \\
&={\overline{\lim}}_{n \to \infty}  \left( \frac{ (2n-1)!!}{n!}\right)^{\frac{1}{n}}  \cdot g(h) \notag \\
&\leq {\overline{\lim}}_{n \to \infty} \left( \frac{ (2n-1)!!}{n!}\right)^{\frac{1}{n}}  \cdot \frac{r}{2} {\label{E:equation_chain}}.
\end{align}
Rewriting $\displaystyle \frac{ (2n-1)!!}{n!} = \frac{ (2n-1)!}{2^{n-1}(n-1)! n!}$ and using 
Stirling's approximation, $n! \sim n^n e^{-n}\sqrt{2\pi n}$ as $n \to \infty$, we obtain
$\displaystyle \lim_{n \to \infty} \left(\frac{ (2n-1)!!}{n!}\right)^{\frac{1}{n}}  =2$, so that
(\ref{E:equation_chain}) reduces to $r$,  
thereby completing the first step of the proof.

\newpar
Now consider the limit
	\[\lim_{h \to 0} \left( \frac{2n+1}{2h} \int_{t=-h}^{t=h} f(t)
	P_n(t/h)\,dt \right) P_n(z/h) ,\]
which may be rewritten as
	\[ \lim_{h \to 0} \left( \frac{2n+1}{2} \int_{t=-1}^{t=1} f(th)
	P_n(t)\,dt \right) P_n(z/h).\]
For each $n$, an application of Taylor's Theorem yields
	\[ f(th)=f(0)+f'(0)th+\frac{f''(0)}{2}(th)^2+ \ldots + \frac{f^{(n)}(0)}{n!} (th)^n+ 	
	\frac{f^{(n+1)}(c_n)}{(n+1)!}(th)^{n+1} \]
for some $c_n$ between $0$ and $t$.

\newpar
For each $0 \leq m \leq n-1$, $t^m$ is orthogonal to $P_n$. Also, 
	\[ \int_{t=-1}^{t=1} t^n P_n(t)\,dt = \frac{2^{n+1}(n!)^2}{(2n+1)!}.\]
(See \cite{Whittaker}.) Thus 
	\begin{equation}{\label{E:taylor1}} 
	\frac{2n+1}{2} \int_{t=-1}^{t=1} f(th)
	P_n(t)\,dt = h^n \cdot \frac{2^{n}(n!)^2}{(2n)!}\cdot \frac{f^{(n)}(0)}{n!} + \mbox{O}(h^{n+1}). 
	\end{equation}
On the other hand, 	
	\begin{equation}{\label{E:taylor2}}
	 P_n(z/h) = \frac{1}{h^n}\left( \frac{(2n)!}{2^n (n!)^2}\cdot z^n + \mbox{O}(h)\right). 
	\end{equation}
Combining (\ref{E:taylor1}) and (\ref{E:taylor2}), we arrive at
\[\lim_{h \to 0^+} \left( \frac{2n+1}{2h} \int_{t=-h}^{t=h} f(t)
	P_n(t/h)\,dt \right) P_n(z/h) =\frac{f^{(n)}(0)}{n!} , \]
which completes the proof of the theorem.
\end{proof}

{\small 
\paragraph*{Acknowledgment.} 
The author wishes to express his gratitude to Paul Nevai for his suggestion that simplified this proof.
}


\clearpage
\begin{thebibliography}{99}
\bibitem{Askey}
R. Askey and D. Haimo, Similarities Between Fourier and Power
Series,
\emph{The American Mathematical Monthly}, {\bf{103}}, (1996), 297-304.

%\bibitem{Borwein}
%Peter Borwein and Tam\'{a}s Erd\'{e}lyi,
%\emph{Polynomials and Polynomial Inequalities}
%Springer-Verlag, Berlin, 1995.

%\bibitem{Dubinsky}
%D. S. Dubinsky,
%Asymptotics of Orthogonal Polynomials: Some Old, Some New, Some Identities,
%\emph{Acta Applicandae Mathematicae}, 61, (2000), 207-256.

%PAGE NUMBER
\bibitem{Neumann}
K. Neumann, \"{U}ber die Entwickelung einer Funktion nach den Kugelfunktionen, \emph{Journal f\"{u}r Mathematik}, 1862.

%FULL JOURNAL CITATION 
\bibitem{Picone}
M. Picone, Maggiorazione di un polinomio di Legendre e delle derivate in un'ellisse a
quello confocale, \emph{Bollettino dell'Unione Matematica Italiana} (3), {\bf{8}}, 1953, 237-242.

\bibitem{Sansone}
G. Sansone,
\emph{Orthogonal Functions},
Interscience Publishers, New York, 1959.

\bibitem{Szego}
Gabor Szeg\"{o},
\emph{Orthogonal Polynomials},
The American Mathematical Society, Providence, 1967.

\bibitem{Whittaker}
E. T. Whittaker and G. N. Watson,
\emph{A Course in Modern Analysis},
Cambridge University Press, London, 1963.



\end{thebibliography}


\end{document}

