The Wigner transformation associated with the Hankel multidimensional operator

Abstract

We consider the Hankel multidimensional operator Δ α , α = ( α 1 , … , α n ) ∈ ] − 1 2 , + ∞ [ n , defined on ] 0 , + ∞ [ n by

We define and study the Wigner transformation V g (called also Gabor transform), where g ∈ L 2 ⁢ ( d ⁢ μ α ) and d ⁢ μ α is the measure defined on [ 0 , + ∞ [ n by

Using harmonic analysis related to the Hankel operator Δ α , we prove a Plancherel theorem and an orthogonality property for the transformation V g . Next, we establish a reconstruction formula for V g and give some applications. In the second part of this work, as applications of the Wigner transformation V g , we define and study the anti-Wick operators A g 1 , g 2 ⁢ ( σ ) , where g 1 , g 2 ∈ L 2 ⁢ ( d ⁢ μ α ) are called window functions and σ ∈ L p ⁢ ( d ⁢ μ α ⊗ d ⁢ μ α ) is a signal. Building on the properties of the Wigner transformation V g , we prove that the operators A g 1 , g 2 ⁢ ( σ ) are bounded linear operators and compact on the Hilbert space L 2 ⁢ ( d ⁢ μ α ) . Finally, we establish a formula of the trace for the anti-Wick operator A g 1 , g 2 ⁢ ( σ ) when the signal 𝜎 belongs to L 1 ⁢ ( d ⁢ μ α ⊗ d ⁢ μ α ) .