Certain isometries related to the bilateral laplace transform

Abstract

We study certain isometries between Hilbert spaces, which are generated by the bilateral Laplace transform

In particular, we construct an analog of the Bargmann‐Fock type reproducing kernel Hilbert space related to this transformation. It is shown that under some integra‐bility conditions on $ the Laplace transform F_F(z), z = x + iy is an entire function belonging to this space. The corresponding isometrical identities, representations of norms, analogs of the Paley‐Wiener and Plancherel's theorems are established. As an application this approach drives us to a different type of real inversion formulas for the bilateral Laplace transform in the mean convergence sense.

Key words:

• bilateral Laplace transform
• Hilbert space
• Sobolev space
• real inversion formula
• Fourier transform
• Hermite polynomials
• Bargmann transform
• Plancherel theorem