Nadia Sraieb

Among the class of generalized Fourier transformations, the linear canonical transform is of pivotal importance mainly due to its higher degrees of freedom in lieu of the conventional Fourier and fractional Fourier transforms. This article is a continuation of our recent work "Linear canonical deformed Hankel transform and the associated uncertainty principles, J. Pseudo-Differ. Oper. Appl.(2023), 14:29". Building upon this, we formulate the generalized translation and convolution operators associated with this newly proposed transformation. Besides, the obtained results are invoked to examine and obtain an analytical solution of the generalized heat equation. Finally, we study the heat semigroup pertaining to the generalized heat equation.

This paper delves into the Dunkl-Bessel operator on and its corresponding harmonic analysis. A generalized form of Heisenberg-type uncertainty inequality is established. Schatten-von Neumann properties for the two-wavelet multiplier within the Dunkl-Bessel theory framework are elucidated. Additionally, the trace formula for a two-wavelet Dunkl-Bessel multiplier is proven as a bounded linear operator in the trace class from into . Furthermore, subject to appropriate conditions, the boundedness and compactness of these Dunkl-Bessel two-wavelet multipliers are proven, applicable to , 1 ≤ p ≤ ∞. Finally, using a class of concentration operators for the Dunkl-Bessel two-wavelet, we show that the eigenfunctions of the Dunkl-Bessel two-wavelet are maximally concentrated in the time-frequency domain. Leveraging this result, we derive approximation inequalities for functions that exhibit significant concentration within specific regions of the time-frequency plane.

Time–frequency (or space–phase) analysis plays a key role in signal analysis. In particular, signals that have a very concentrated time–frequency content are of great importance. However, the uncertainty principle sets a limitation to the possible simultaneous concentration of a function and its Dunkl–Bessel transform. For this purpose, we introduce and study a new transformation called Dunkl–Bessel Gabor transform. For this transformation, we define the Toeplitz-type (or time–frequency localization) operators, in order to localize signals on the time–frequency plane. We study these operators; in particular, we give criteria for their boundedness and Schatten class properties. Then, using the special class of concentration operators, which are compact and self-adjoint, we show that their eigenfunctions are maximally time–frequency-concentrated in the region of interest.