TITLE: Are the Hermite-Gauss functions linearly dense in $L^1(\mathbb{R})$?
QUESTION [7 upvotes]: The Hermite-Gauss functions ($t\mapsto H_m(t)e^{-t^2/2}$) are known to be an orthonormal basis for $L^2(\mathbb{R})$, a fortiori linearly dense in $L^2(\mathbb{R})$, and all are in the Schwartz space (and hence in $L^p(\mathbb{R})$). 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 $L^1(\mathbb{R})$ - and, more generally, $L^p(\mathbb{R})$. 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 $L^p(\mathbb{R})$ 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?

REPLY [3 votes]: Hermite functions are dense in $L^1[\mathbb{R}]$, but we cannot use coefficients computed as above. Every $L^1[\mathbb{R}]$ function can be approximated in $L^1[\mathbb{R}]$ by continuous bounded functions of compact support - so they are also in $L^2[\mathbb{R}]$ and can be approximated by linear combinations of Hermite functions.