Convergence of orthogonal polynomial expansions

Question

"Everyone" knows that for a general $f\in L^2[0,1]$, the Fourier series of $f$ converges to $f$ in the $L^2$ norm but not necessarily in most other senses one might be interested in; but if $f$ is reasonably nice, then its Fourier series converges to $f$, say, uniformly.

I'm looking for similar results about orthogonal polynomial expansions for functions on the whole real line. What I specifically want at the moment is sufficient conditions on a bounded function $f:\mathbb{R} \to \mathbb{R}$ so that the partial sums of its Hermite polynomial expansion are uniformly bounded on compact sets, but I'm also interested in learning what's known about pointwise/uniform/etc. convergence results for Hermite and other classical orthogonal polynomials.

Possibly such results follow trivially from well-known basic facts about Hermite polynomials, but I'm not familiar with that literature and I'm having trouble navigating it. So in addition to precise answers, I'd appreciate literature tips (but please don't just tell me to look at Szegő's book unless you have a specific section to recommend).

Answer 1

Define $\psi_n(x) = c_n H_n(x) e^{-x^2/2}$ as in http://en.wikipedia.org/wiki/Hermite_polynomials . Also define the differential operator $H u = - u'' + x^2 u$. Then the $\psi_n$ form an othonormal basis of $L^2$ and $H \psi_n = (2n + 1) \psi_n$.

Warning: As coudy points out below: one needs $\|H f\| < \infty$ and not just $\langle f, Hf\rangle < \infty$. So the computations below need to be changed.

Rest of original post

Given now $f$ such that $$ A = \langle f,Hf \rangle =\int \overline{f(x)} (Hf)(x) dx $$ is finite. Then by writing $f(X) = \sum_{n \geq 0} f_n \psi_n(X)$, we obtain $$ A = \langle f,Hf \rangle =\langle f, \sum_{n \geq 0} f_n H\psi_n(X) \rangle = \langle \sum_{n \geq 0} f_n \psi_n(X) , \sum_{n \geq 0} f_n (2 n + 1)\psi_n(X) \rangle $$ Now using orthonormality of the $psi_n$, we conclude that $$ A = \sum_{n \geq 0} |f_n|^2 (2n + 1). $$ Now using that the $\psi_n(x)$ are all bounded by $2$ it follows that the sequence converges uniformly!

Now, what does $\langle f,Hf \rangle < \infty$ mean for $f(x) = e^{-x^2/2} g(x)$. This can be computed to mean $$ \int |g'(x) + \frac{x}{2} g(x)|^2 e^{-x^2} dx. $$

On a philosophical level, this is not about the $H_n$ being orthogonal polynomials, but about them being eigenfunctions of a self-adjoint operator. (well the $\psi_n$ are).

Answer 2

Let me recall a quick $L^2$ proof of the uniform convergence of the Fourier series of a $C^1$ function $f$. Let $c_n$ be its Fourier coefficients. Then

$$|f(x)| \leq |c_0|+ \Sigma|c_n|\ n \ {1\over n}\ \ \leq |c_0| + \ \ \sqrt{\Sigma \ n^2 |c_n|^2}\ \ \sqrt{\Sigma\ 1/n^2}$$

Replacing f by f minus its partial sum, and noting that $\Sigma \ n^2|c_n|^2 = ||f'||_2^2 \ $ is finite, you get uniform convergence.

So maybe you can use a similar computation in case of a family of orthogonal polynomials ?

Answer 3

I too was looking into this recently and found good citations difficult to find. Here are two I found which I can recommend:

.1. In the book "Special Functions and their Applications" by Lebedev, he gives the following clear theorem statement (with a moderately technical but not too bad-looking proof): (Theorem 2) "Assume $f \in L^2(\gamma)$ (i.e., is square-integrable w.r.t. the standard Gaussian measure) and is piecewise-$\mathcal{C}^1$ on every finite interval $[-a,a]$. Then the Hermite expansion of $f$ converges pointwise at every point of continuity of $f$", and further, (Remark 1) "converges to $(f(x^+)+f(x^-))/2$ at any jump discontinuity $x$."

It would be nice to also know that one has uniform convergence on any interval $[-a,a]$, but I didn't immediately see how to read that out.

.2. The book "Gaussian measures" by Bogachev gives pretty careful statements about the domains under which formal operations (e.g., differentiation of Hermite expansions) hold.