This is an orthonormal basis for . Since the measure of is infinite, functions will have to decay at infinity in order to be in . The Hermite functions are
where is the nth Hermite polynomial, defined by
The goal is to prove that the functions can be obtained from via the Gram-Schmidt process. (They actually form a basis, but I won’t prove that.)
One can observe that the term would be unnecessary if we considered the weighted space with weight and the inner product . In this language, we orthogonalize the sequence of monomials and get the ON basis of polynomials with being a normalizing constant. But since weighted spaces were never introduced in class, I’ll proceed with the original formulation. First, an unnecessary graph of ; the order is red, green, yellow, blue, magenta.
Claim 1. is a polynomial of degree with the leading term . Proof by induction, starting with . Observe that
where the first term has degree and the second . So, their sum has degree exactly , and the leading coefficient is . Claim 1 is proved.
In particular, Claim 1 tells us that the span of the is the same as the span of .
Claim 2. for . We may assume . Must show . Since is a polynomial of degree , it suffices to prove
(*) for integers .
Rewrite (*) as and integrate by parts repeatedly, throwing the derivatives onto until the poor guy can't handle it anymore and dies. No boundary terms appear because decays superexponentially at infinity, easily beating any polynomial factors. Claim 2 is proved.
Combining Claim 1 and Claim 2, we see that belongs to the -dimensional space , and is orthogonal to the -dimensional subspace . Since the “Gram-Schmidtization'' of has the same properties, we conclude that agrees with this “Gram-Schmidtization'' up to a scalar factor.
It remains to prove that the scalar factor is unimodular ( since we are over reals).
Claim 3. for all . To this end we must show . Expand the first factor into monomials, use (*) to kill the degrees less than n, and recall Claim 1 to obtain
As in the proof of Claim 2, we integrate by parts throwing the derivatives onto . After n integrations the result is
, as claimed.
P.S. According to Wikipedia, these are the “physicists’ Hermite polynomials”. The “probabilists’ Hermite polynomials” are normalized to have the leading coefficient 1.