Let be an entire function satisfying
in the upper half-plane. Define a Hilbert space of entire functions
to be the set of all entire functions such that
is square integrable on the real axis and such that
de Branges [ MR 87m:11050] and [ MR 93f:46032] proved the following beautiful
Theorem . Let be an entire function having no real zeros such that for , such that for a constant of absolute value one, and such that is a strictly increasing function of for each fixed real . If for every element with , then the zeros of lie on the line , and when is a zero of .
. Then it can be shown that
for , and that
is a strictly increasing function of on for
each fixed real .
Therefore, it is natural to ask whether the Hilbert
space of entire functions satisfies the condition that
It is shown in [ MR 2001h:11114] that this condition is not satisfied.
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