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 the recent works on the linear canonical Dunkl transforms carried out in Ghazouani et al. (J Math Anal Appl 449:1797–1849, 2017), Mejjaoli (J Pseudo-Differ Oper Appl 16:1–43, 2025). Building upon this, we will introduce and study in this paper the generalized wavelet transform associated with the LCDT, called the Dunkl linear canonical wavelet transform. Then we will formulate several weighted uncertainty principles for this new transformation.

We examine the Sobolev space associated with the linear canonical Dunkl transform and explore some properties of the linear canonical Dunkl operators. Building on these results, we establish a real Paley–Wiener theorem for the linear canonical Dunkl transform. Further, we characterize the square-integrable function f whose linear canonical Dunkl transform of the function is supported in the polynomial domain. Finally, we develop the Boas-type Paley–Wiener theorem for the linear canonical Dunkl transform.

In this paper, we introduce the localization operator associated with the linear canonical continuous Dunkl wavelet transform. We analyze the boundedness of the operator Lψ,φ(σ) for various classes of symbols and wavelet functions. We also establish the compactness of the localization operator on Lkp(R) spaces, where 1≤p≤∞. Additionally, we explore the properties of the localization operator in Schatten-von Neumann classes and demonstrate that, with appropriate choices of symbols and wavelet functions, the localization operator can be identified as both a trace class operator and a Hilbert–Schmidt operator.

The linear canonical Dunkl transform (LCDT) is a novel addition to the class of linear canonical transforms, which has gained a respectable status in the realm of harmonic analysis within a short span of time. Knowing the fact that the study of the theory of the wavelet multipliers is both theoretically interesting and practically useful, we investigate this theory for the LCDT. First, we introduce the notion of the linear canonical Dunkl multiplier and examine the underlying theory of the two-wavelet linear canonical Dunkl multiplier operator. Particularly, we study the trace class properties of such operators and also demonstrate that they belong to the Schatten-von Neumann class. Second, special attention is paid to the boundedness and compactness of the proposed operators on Lkp(rbhiN), 1 ≤ p ≤∞. We culminate our study by formulating several typical examples of the two-wavelet LCDMO and some applications.

In this work, we define the composition of wavelet transforms and obtain its Parsevals's identity. Furthermore, we discuss the convolution operator and continuous Dunkl wavelet transform as time-invariant filters. The physical interpretation and potential application of time-invariant filter involving Fredholm type integral are obtained.

In this paper, the dual convolution structure for the Mehler-Fock transform is defined and obtained its estimates on Lebesgue space. Next, the two symbols σ(x, τ) and ρ(y, τ1) as an inverse Mehler-Fock transform of some measurable functions are introduced and defined the two pseudo-differential operators Pσ and Qρ respectively. Moreover, the product of two pseudo-differential operators is shown as a pseudo-differential operator. Furthermore, the boundedness of this operator in Sobolev-type space by using the dual convolution is shown and discussed the some special cases.

This paper aims to study the theory of variation diminishing convolution kernel in the fractional Hankel-type transform domain. Moreover, necessary and sufficient conditions are established for the function Gθ to be the variation diminishing convolution kernel.

The continuous wavelet transform (CWT) associated with the index Whittaker transform is defined and discussed using its convolution theory. Existence theorem and reconstruction formula for CWT are obtained. Moreover, composition of CWT is discussed, and its Plancherel's and Parseval's relations are also derived. Further, the discrete version of this wavelet transform and its reconstruction formula are given. Furthermore, certain properties of the discrete Whittaker wavelet transform are discussed.

The main objective of this paper is to study the composition of continuous Kontorovich-Lebedev wavelet transform (KL-wavelet transform) and wave packet transform (WPT) based on the Kontorovich-Lebedev transform (KL-transform). Estimates for KL-wavelet and KL-wavelet transform are obtained, and Plancherel’s relation for composition of KL-wavelet transform and WPT-transform are derived. Recon-struction formula for WPT associated to KL-transform is also deduced and at the end Calderon’s formula related to KL-transform using its convolution property is obtained.

In this paper, we define the windowed-Mehler–Fock transform and introduce the corresponding Weyl transform. Further, we examine the boundedness of windowed-Mehler–Fock transform in Lebesgue space and establish some of its fundamental properties. Also, we give the criteria of boundedness and compactness of Weyl transform in Lebesgue space.

The main aim of this paper is to study the notion of the heat kernel associated with the zero order Mehler–Fock transform. The upper bound of the heat kernel is obtained. We discuss its some properties and fundamental solution of a generalized diffusion equation. Weierstrass type integral transform with the heat kernel is established. Further boundedness of the Weierstrass integral transform on Sobolev space is discussed and obtained its inversion formula also. Moreover we obtain Heisenberg type inequality associated with the Mehler–Fock transform and heat kernel.

In this article, we introduce a new index transform associated with the cone function Pi √τ− 1 2 (2√x), named as Mehler-Fock-Clifford transform and study its some basic properties. Convolution and translation operators are defined and obtained their estimates under Lp (I; x− 12 dx) norm. The test function spaces Gα and Fα are introduced and discussed the continuity of the differential operator and MFC-transform on these spaces. Moreover, the pseudo-differential operator (p.d.o.) involving MFC-transform is defined and studied its continuity between Gα and Fα.

The present paper is devoted to the study of the Mehler-Fock transform with index as the Legendre function of first kind. Continuity property of the Mehler-fock transform on the test function spaces Λ α and G α is given. Moreover pseudo-differential operator (p.d.o.) with symbol σ (x, τ) ∈ S m in terms of Mehler-Fock transform is defined and also its continuity property from test function space G α into Λ α is shown. The Mehler-Fock potential (MF-potential) P s σ is defined on G α (I) space and it is extended to the space of distribution. Also some properties of MF-potential are discussed. At the end Sobolev type space V s,p (I) is defined and it is shown that MF-potential is an isometry of V s,p (I).

The continuous wavelet transform (CWT) associated with zero-order Mehler-Fock transform (MF-transform) is defined and discussed its some basic properties, Plancherel's and Parseval's relations, reconstruction formula for CWT are obtained. Further composition of CWT is investigated and then its Parseval's and Plancherel's relations are given. Moreover, time-invariant filter has been defined and proved convolution operator and wavelet transform are represented as time-invariant transform.

The main aim of this paper is to find out the estimates of convolution for zero-order Mehler–Fock transform with various approaches. Pseudo-differential operator in terms of zero-order Mehler–Fock transform is defined and obtained its another integral representation. Further, its estimate in Lebesgue space has been studied. At the end some applications of zero-order Mehler–Fock transform and of convolution are discussed.