Vtyjkyuk

The Hermite-Gauss functions ($tmapsto H_m(t)e^-t^2/2$) are known to be an orthonormal basis for $L^2(mathbbR)$, a fortiori linearly dense in $L^2(mathbbR)$, and all are in the Schwartz space (and hence in $L^p(mathbbR)$). These functions play an extremely important role in $L^2$ theory, Fourier transform theory, quantum mechanics, and elsewhere. I've never seen it mentioned whether or not the Hermite-Gauss functions are linearly dense in $L^1(mathbbR)$ - and, more generally, $L^p(mathbbR)$. It seems like a reasonable question to ask but Googling has not led me to an answer one way or another. My guess is that they are, in fact, not linearly dense in $L^p(mathbbR)$ except for $p=2$, but I don't have any clue as to how to prove that. Can anyone point me to this result or maybe give a clue as to why it is or isn't true?

The functions $g_m(t) = H_m(t),e^-t^2/2 $ just give an orthogonal base of $L^2(mathbbR)$, since:
$$ int_-infty^+infty f_m(t)^2,dt = 2^m m! sqrtpi. $$
An orthonormal base is given by:
$$ f_m(x) = frac1sqrt2^n n! sqrtpi,H_m(x), e^-x^2/2. $$
We may notice that if $m$ is odd then $int_mathbbRf_m(x),dx = 0$, while:
$$ int_mathbbR f_2n(x),dx = frac(2n)!n!sqrt2pifrac1sqrt4^n (2n)! sqrtpi=sqrtfrac2sqrtpi4^nbinom2nnapprox sqrt2cdot n^-1/4.$$
Moreover, we have $left|,f_m(x),right|leqfrac12$ for every $m$ and every $x$, and $f_m$ is essentially zero outside $left[-fracpi2sqrtm,fracpi2sqrtmright]$. So we know the behaviour of our base with respect to $L^1,L^2,L^infty$. By interpolation or other techniques, it is not difficult to check that linear combinations of $f_n$s cannot be dense in $L^1$.

For instance, we may consider the function $g(x)=frac1sqrtxcdotmathbb1_(0,1)(x)$ that belongs to $L^1setminus L^2$.

Every $f_n$ has a rather large support, so if we compute:
$$ a_n = int_0^1 g(x),f_n(x),dx $$
we have that:
$$ g_N(x)=sum_n=0^N a_n,f_n(x) $$
is a not-so-bad approximation of $g(x)$ over $[0,1]$, but it has a long tail, such that $g_N(x)$ does not converge to $g(x)$ in $L^1(mathbbR)$. To make this argument visually appealing, this is the situation for $N=10$:

$hspace0.5in$

This non-density argument is even more convincing (at least to me) if we notice that every $g_N$ is an entire function (of order 2) while $g$ is a compact-supported function with a branch-like singularity in a right neighbourhood of the origin. So the fact that $L^2$ and the Schwarz space are mapped into theirselves by the Fourier transform is really crucial for proving the density of the span of the "Hermite base".

The ordinary heat equation is
$$
fracpartial Fpartial t=fracpartial^2Fpartial x^2,\
F(0,x)=f(x).
$$
The initial heat distribution is $f$. The ordinary heat kernel is the Guassian, and the resulting time evolution operator $T(t)=e^tL$ is a constractive $C_0$ semigroup on every $L^p(mathbbR)$ for $1 le p < infty$; the fact that it is $C_0$ gives
$$
|e^tLf-f|_prightarrow 0 mbox as tdownarrow 0,;; 1 le p < infty.
$$
That gives a nice approximation. In fact, this approximation technique goes back to Weierstrass in his original proof of the Weierstrass Approximation Theorem.

The Hermite functions $h_n(x)=H_n(x)e^-x^2/2$ are the $L^2$ eigenfunctions of
$$
Lf = -fracd^2fdx^2+x^2f
$$
with eigenvalues $lambda = 2n+1$ for $n=0,1,2,3,cdots$. The Hermite functions $ h_n _n=0^infty$ form a complete orthonormal basis of $L^2(mathbbR)$. The heat equation associated with $L$ is
$$
fracpartial Fpartial t=fracpartial^2Fpartial^2x-x^2F,\
F(0,x)=f(x).
$$
This heat equation is better behaved in many ways than the ordinary heat equation because $-x^2F$ pulls heat out of the system near $pminfty$ for positive $F$. The time evolution solution operator $T(t)=e^tL$ in this case is
$$
T(t)f = sum_n=0^inftye^-(2n+1)t(f,h_n)h_n.
$$
So the approximation problem by Hermite functions is closely related to the continuity properties of the heat solution at $t=0$. That's why one studies the Hermite kernel function
$$
K(r,x,y)=sum_n=0^inftyr^nh_n(x)h_n(y),
$$
which has an explicit representation as a bivariate Gaussian:
$$
K(r,x,y) = frac1sqrtpi(1-r^2)
expleft-frac14frac1-r1+r(x+y)^2-frac14frac1+r1-r(x-y)^2right.
$$
So the approximation problem can be studied by looking at the question
$$
f;; ? = ?;; lim_rdownarrow 0int_-infty^inftyK(r,x,y)f(y)dy = lim_rdownarrow 0sum_n=0^inftyr^n(f,h_n)h_n(x).
$$

Hermite functions are dense in $L^1[mathbbR]$, but we cannot use coefficients computed as above. Every $L^1[mathbbR]$ function can be approximated in $L^1[mathbbR]$ by continuous bounded functions of compact support - so they are also in $L^2[mathbbR]$ and can be approximated by linear combinations of Hermite functions.