In this paper, we present the basic k-Hankel wavelet theory. Next, we study the boundedness and compactness of localization operators associated with k-Hankel wavelet transforms on $L_k^p\left(\mathbb{R}\right)$, $1\le p\le \mathrm{\infty }$. Finally, we give some typical examples of localization operators.

The deformed wavelet transform is a new addition to the class of wavelet transforms that heavily rely on a pair of generalized translation and dilation operators governed by the well-known Dunkl transform. In this study, we adapt the symmetrical properties of the Dunkl Laplacian operator to prove a class of quantitative uncertainty principles associated with the deformed wavelet transform, including Heisenberg’s uncertainty principle, the Benedick–Amrein–Berthier uncertainty principle, and the logarithmic uncertainty inequalities. Moreover, using the symmetry between a square integrable function and its Dunkl transform, we establish certain local-type uncertainty principles involving the mean dispersion theorems for the deformed wavelet transform.

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 $L^p$ 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.

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 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.