May 27, 2016

Xing-Tang Dong, Kehe Zhu

# Introduction

The Fock space F2 is the Hilbert space encompassing all entire functions f, existing on the complex plane C, with a Gaussian measure. F2 is frequently used in various problems in functional analysis, mathematical physics, and engineering. On the other hand, we acknowledge another Hilbert space as L2(R) = L2(R, dx). Here, the focus is on studying the (fractional) Fourier transform, the (fractional) Hilbert transform, and the wavelet transform as bounded linear operators on L2(R).

The Bargmann transform B is a crucial operator from L2(R) to F2 and presents itself as an intrinsic unitary operator from L2(R) to F2. Interestingly, the reverse of B is also an integral operator. This transform is a long-established tool combining mathematical analysis and mathematical physics. We explore the consequences of the Bargmann operation on a myriad array of classical integral operators on L2(R).

# Preliminaries

To introduce an orthonormal basis for L2(R), it is crucial to consider Hn(x), the nth Hermite polynomial. A result of this, the Bargmann transform is acknowledged as a unitary operator from L2(R) to F2. Further, it's known that a, b∈R with a >0 implies that R e −(a+ib)(x+z)2 dx= √ π / √ a+ib, valid for any complex number z.

# The Fourier Transform

The Fourier transform is primarily defined by F(f)(x) = 1/√π ∫R e−2ixt f(t)dt. Understandably, the Fourier transform is a unitary operator on L2(R) and its inverse is rendered as F−1(f)(x) = 1/ √π ∫R e2ixt f(t)dt. Considering this, the fractional Fourier transforms become one of the most influential tools in mathematics, quantum mechanics, optics, and signal processing. Therefore, for any real angle α, we define the α-angle fractional Fourier transform. The α-angle fractional Fourier transform morphs into the default Fourier transform and the inverse Fourier transform.

Also, what remains noteworthy is that under the Bargmann transform, the operator Fα: L2(R) → L2(R) is unitarily similar to an extremely convenient operator on the Fock space F2. In the end, The operator T=BFαB−1: F2 → F2 applies and gives Tf(z) =f(e−iαz) for all f∈F2.

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