Square integrable real entire functions form a Hilbert space?

Asked Nov 25 '21 at 18:50

Active Nov 25 '21 at 18:50

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I am interested in the space of real entire functions $f(x)$ which are square integrable, i.e. in $L^2((-\infty,\infty))$. By real entire, I mean its complex extension $f(z)$ is entire. Does this space form a Hilbert space, with $L^2$ inner product?

This appears similar to a Bergman space $A^2(D)$, except we are considering entire functions (analytic not only on $D$ but the entire complex plane), and simultaneously restricting the domain to the real line. Of course, $A^2(\mathbb{C})$ is trivial, as there are no nonzero square integrable entire functions on this domain.

asked Nov 25 '21 at 18:50

fewfew4

For example, your space contains $\exp(-a (x-b)^2)$ for any $a > 0$. I would expect that the linear span of these is dense in $L^2$. In particular it is not closed. – Robert Israel Nov 25 '21 at 20:00
@RobertIsrael would you expect the space of square integrable real analytic functions to be closed? – fewfew4 Nov 25 '21 at 20:24
No, "it" in my comment refers to your space. – Robert Israel Nov 25 '21 at 21:42

Square integrable real entire functions form a Hilbert space?

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