Math 6420 • PDE

Prove that the Hermite polynomials form a complete set on
\(-\infty<x<\infty\) with the inner product
\(\langle u,v\rangle=\int_{-\infty}^{\infty}u(x)v(x)e^{-x^2/2}\,dx\).

Hint: The following steps will prove useful:
\begin{enumerate}
\item[(a)] Show that if the Hermite-Bessel coefficients of \(f(x)\) are all zero, then
\(\int_{-\infty}^{\infty}e^{-x^2/2}x^nf(x)\,dx=0\) for all \(n\).
\item[(b)] Show that the Fourier transform \(F(z)\) of the function \(f(x)e^{-x^2/2}\) has no
singularities in the complex plane and is therefore an entire function.
\item[(c)] Calculate the Taylor series of \(F(z)\) and show that it is zero. What conclusion
can you draw about \(f(x)\) and what does this say about Hermite polynomials?
\end{enumerate}