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The deformed Hankel wavelet transform (( k ,  n )-HWT) is a novel addition to the class of wavelet 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 localization operators is both theoretically interesting and practically useful, we investigated several subjects of time-frequency analysis for the deformed Hankel wavelet transform. First, we study the L p boundedness and compactness of localization operators associated with the deformed Hankel wavelet transforms on R . Next, involving the reproducing kernel and spectral theories we investigate the time-frequency operators. Finally, the scalogram for the deformed Hankel wavelet transform is introduced and studied at the end.

The k -Hankel wavelet transform ( k -HWT) is a novel addition to the class of wavelet 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 theory of the localization operators is both theoretically interesting and practically useful, we investigate this theory for the k -HWT. Firstly, we study the L p boundedness governing the simultaneous localization of a signal and the corresponding k -HWT. Secondly, we investigate the L p compactness of localization operators associated with the k -HWT. We culminate our study by formulating several typical examples of localization operators associated with the k -HWT.

We introduce the notion of localization operators associated with the Riemann-Liouville-Wigner transform, and we give a trace formula for the localization operators associated with the Riemann-Liouville-Wigner transform as a bounded linear operator in the trace class from 𝐿²(𝑑𝜈𝛼 ) into 𝐿²(𝑑𝑣𝛼 ) in terms of the symbol and the two admissible wavelets. Next, we give results on the boundedness and compactness of localization operators associated with the Riemann-Liouville-Wigner transform on 𝐿𝑝(𝑑𝜈𝛼 ), 1 ≤ 𝑝 ≤ ∞.

We prove a new inversion formula for the Dunkl Gabor transform. We also prove several versions of Heisenberg-type inequalities, Donoho–Stark’s uncertainty principles, local Cowling–Price’s type inequalities and finally Faris–Price’s uncertainty principle for the previous transform.

We consider the continuous wavelet transform Φ h W associated with the Heckman–Opdam operators on R d . We analyse the concentration of this transform on sets of finite measure. In particular, Donoho–Stark and Benedicks-type uncertainty principles are given. Next, we prove many versions of Heisenberg-type uncertainty principles for Φ h W . Finally, we investigate the localization operators for Φ h W , in particular we prove that they are in the Schatten–von Neumann class.

We prove a new inversion formula for the Dunkl Gabor transform. We also prove several versions of Heisenberg-type inequalities, Donoho–Stark’s uncertainty principles, local Cowling–Price’s type inequalities and finally Faris–Price’s uncertainty principle for the previous transform.

In this paper, we establish real Paley-Wiener theorems for the hypergeometric Fourier transform on [InlineEquation not available: see fulltext.]. More precisely, we characterize the functions of the generalized Schwartz space [InlineEquation not available: see fulltext.] and of [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], whose hypergeometric Fourier transform has bounded, unbounded, convex, and nonconvex support. Finally we study the spectral problem on the generalized tempered distributions [InlineEquation not available: see fulltext.]. MSC: 35L05, 22E30.