TITLE: Convergence of orthogonal polynomial expansions
QUESTION [6 upvotes]: "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).

REPLY [5 votes]: 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.