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The Whittaker Gabor transform (WGT) is a novel addition to the class of Gabor transforms, which has gained a respectable status in the realm of time-frequency signal analysis within a short span of time. Knowing the fact that the study of the time-frequency analysis is both theoretically interesting and practically useful, the aim of this article is to explore two more aspects of the time-frequency analysis associated with the WGT including the spectral analysis associated with the concentration operators and the scalogram.

We introduce the notion of the ( k ,  a )-generalized wavelet transform. Particular cases of such generalized wavelet transform are the classical and the Dunkl wavelet transforms. The restriction of the ( k ,  a )-generalized wavelet transform to radial functions is given by the generalized Hankel wavelet transform. We prove for this new transform Plancherel’s formula, inversion theorem and a Calderón reproducing formula. As applications on the ( k ,  a )-generalized wavelet transform, we give some applications of the theory of reproducing kernels to the Tikhonov regularization on the generalized Sobolev spaces. Next, we study the generalized wavelet localization operators.

In this paper, we consider the deformed Stockwell transform. Knowing the fact that the study of the time-frequency analysis are both theoretically interesting and practically useful, we investigated several problems for this subject on the setting of the deformed Stockwell transform. Firstly, we study the boundedness and compactness of localization operators associated with the deformed Stockwell transforms on R . Next, we present typical examples of localization operators. Finally, the scalogram for the deformed Stockwell transform are introduced and studied at the end.

One of the best known time–frequency tools for examining non-transient signals is the linear canonical windowed transform, which has been used extensively in signal processing and related domains. In this paper, by involving the harmonic analysis for the linear canonical Dunkl transform, we introduce and then study the linear canonical Dunkl windowed transform (LCDWT). Given that localization operators are both theoretically and practically relevant, we will focus in this paper on a number of time–frequency analysis topics for the LCDWT, such as the Lp boundedness and compactness of localization operators for the LCWGT. Then, we study their trace class characterization and show that they are in the Schatten–von Neumann classes. Then, we study their spectral properties in order to give some results on the spectrograms for the LCDWT.

In the present paper we define and study a new wavelet transformation associated to the linear canonical Dunkl transform (LCDT), which has been widely used in signal processing and other related fields. Then we define and study a class of pseudo-differential operators known as time-frequency (or localization) operators and we give criteria for its boundedness and Schatten class properties.

The aim of this paper is to show some uncertainty inequalities for the linear canonical Dunkl transform (LCDT), including sharp Heisenberg-type, entropic-type, logarithmic-type, Donoho–Stark-type and local-type uncertainty principles.

Among the class of generalized Fourier transformations, the linear canonical transform is of crucial importance, mainly due to its higher degrees of freedom compared to the conventional Fourier and fractional Fourier transforms. In this paper, we will introduce and study two versions of wavelet transforms associated with the linear canonical Dunkl transform. More precisely, we investigate some applications for Dunkl linear canonical wavelet transforms. Next we will introduce and develop the harmonic analysis associated with the Dunkl linear canonical wavelet packets transform. We introduce and study three types of wavelet packets along with their associated wavelet transforms. For each of these transforms, we establish a Plancherel and a reconstruction formula, and we analyze the associated scale-discrete scaling functions.

In the present paper we study a class of Toeplitz operators called concentration operators that are self-adjoint and compact in the linear canonical Dunkl setting. We show that a finite vector space spanned by the first eigenfunctions of such operators is of a maximal phase-space concentration and has the best phase-space concentrated scalogram inside the region of interest. Then, using these eigenfunctions, we can effectively approximate functions that are essentially localized in specific regions, and corresponding error estimates are given. These research results cover in particular the classical and the Hankel settings, and have potential application values in fields such as signal processing and quantum physics, providing a new theoretical basis for relevant research.

The linear canonical Fourier transform is one of the most celebrated time-frequency tools for analyzing non-transient signals. In this paper, we will introduce and study the deformed Gabor transform associated with the linear canonical Dunkl transform (LCDT). Then, we will formulate several weighted uncertainty principles for the resulting integral transform, called the linear canonical Dunkl-Gabor transform (LCDGT). More precisely, we will prove some variations in Heisenberg’s uncertainty inequality. Then, we will show an analog of Pitt’s inequality for the LCDGT and formulate a Beckner-type uncertainty inequality via two approaches. Finally, we will derive a Benedicks-type uncertainty principle for the LCDGT, which shows the impossibility of a non-trivial function and its LCDGT to both be supported on sets of finite measure. As a side result, we will prove local uncertainty principles for the LCDGT.

In the present article, we prove new quantitative uncertainty principles for the directional short-time Fourier transform. Next, we introduce the notion of the generalized wavelet multipliers associated with the inverse of the directional short-time Fourier transform. We study the boundedness, Schatten class properties of these operator and give a trace formula. In particular we prove that the generalized Landau-Pollak-Slepian operator is a generalized wavelet multiplier. Finally, we investigate the boundedness and compactness of the generalized wavelet multipliers in the L p -spaces.