The Segal-Bargmann transform and the Segal-Bargmann space
1 The Fourier transform
Let . For Borel measurable functions , when is integrable we define
For , for almost all , is integrable,11 1 Walter Rudin, Real and Complex Analysis, third ed., p. 170, Theorem 8.14. and using Fubini’s theorem one checks that
Let be the Schwartz functions . For a multi-index and define by
One proves that
Parseval’s formula states that for ,
For , using Cauchy’s integral theorem we obtain
2 The heat kernel
For and , define by
and we calculate
The Fourier transform of is22 2 http://individual.utoronto.ca/jordanbell/notes/stationaryphase.pdf, Theorem 2.
Using and the Fourier inversion theorem,
For and for ,
It is apparent that is holomorphic. By the dominated convergence theorem,
and is holomorphic.
3 The Segal-Bargmann transform and the Segal-Bargmann space
Let be Lebesgue measure on , for let
and let be the Borel measure on whose density with respect to is . We define to be the set of those holomorphic functions satisfying
and for we define
We call the Segal-Bargmann space. It can be proved that it is a Hilbert space.
For write , and applying Parseval’s formula and (1) yields
Using this with and then using Fubini’s theorem and an identity for Gaussian integrals33 3 http://individual.utoronto.ca/jordanbell/notes/stationaryphase.pdf, Theorem 3. we get
Therefore is a linear isometry. We call the Segal-Bargmann transform. It can be proved that is a Hilbert space isomorphism.44 4 cf. https://www.math.lsu.edu/~olafsson/pdf_files/ht.pdf
For and , write
For and let , and for , using
Then is a reproducing kernel for the Hilbert space .