Frame theory is recently an active research area in mathematics, computer science, and engineering with many exciting applications in a variety of different fields.This theory has been generalized rapidly and various generalizations of frames in Hilbert spaces In this papers we study the notion of dual continuous K-frames in Hilbert spaces. Also we etablish some new properties.

Two estimates useful in applications are proved for the generalized Fourier-Bessel transform in the space L 2 α,n as applied to some classes of functions characterized by a generalized modulus of continuity.

The main purpose of this paper is to define the Wigner transform associated with the Jacobi-Dunkl operator and to give some results related to this transform as inversion formula, next motivated by Wong’s point of view we define a class of pseudo-differential operator Lu,v(σ) called localization operator wich depend on a symbol σ and two functions u and v, we give a criteria in terms of the symbol σ for its boundedness and compactness,we also show that these operators belongs to the Schatten-Von Neumann class Sp for all p ∈ [1; +∞] and we give a trace formula. MSC2020-Mathematics Subject Classification: 42B10, 47G30, 47B10.

The main crux of this work is to study the Wigner transform associated with the Laguerre-Bessel operator and to give some results related to this transform. Next motivated by Wong's point of view we define a class of pseudo-differential operator Lu,v (σ) called localization operator wich depend on a symbol σ and two functions u and v, we give a criteria in terms of the symbol σ for its boundedness and compactness, we also show that these operators belongs to the Schatten-Von Neumann class S p for all p ∈ [1; + ∞] and we give a trace formula. MSC2020-Mathematics Subject Classification: 42B10, 47G30, 47B10.

Note: Please see pdf for full abstract with equations. The aim of this paper is to introduce a translation operator associated to the generalized linear canonical Fourier Bessel transform Fmα,n and study some of the important properties, and we derive a convolution product for this transform and as application we study the heat equation and the heat semigroup related to the conjugate of the generalized Bessel operator Δmα,n. MSC Classification: 33D15; 47A05; 47B90