In this paper, we define and study the Gabor transform in the context of the Laguerre hypergroup. We prove some of its basic properties, such as Plancherel theorem, inversion formula and Calder´on’s reproducing inversion formula. Next, using the harmonic analysis related to Laguerre hypergroup, we examine spaces of Sobolev type for which we make explicit kernels reproducing. Exploiting the aforesaid theory, we introduce and study the extremal function associated with the Gabor transform. Finally, by utilizing the reproducing kernels we establishimportant estimates for this extremal function.

ABSTRACTThe three‐dimensional octonionic hyperbolic Fourier transform (OHFT) is introduced as a hypercomplex integral Fourier transform associated with three‐dimensional octonion‐valued signals defined within an open rectangular prism , where , in Euclidean space , equipped with a hyperbolic measure. This new transform reduces to the octonion Fourier transform when and tend to . The purpose of this paper is to define the OHFT and establish its fundamental properties, including linearity, hyperbolic modulation, scaling, reflection, the inversion formula, Plancherel's theorem, Parseval's formula, the Hausdorff–Young inequality, and partial derivatives for three‐dimensional hyperbolic octonion functions. I also demonstrate the application of the OHFT in solving two linear partial differential equations within the octonionic hyperbolic framework and extend the Heisenberg–Weyl uncertainty principle to the OHFT domain.

We construct explicitly a Kac–Moody algebra associated to SL[Formula: see text] in two different but equivalent ways: either by identifying a Hilbert basis of [Formula: see text] or by the Plancherel Theorem. Central extensions and Hermitian differential operators are identified.

We define and study the Stockwell transform \(\mathscr{S}_g\) associated with the Whittaker operator \[\Delta_{\alpha}:=-\frac{1}{4}\left[x^2\frac{\mbox{d}^2}{\mbox{d}x^2}+(x^{-1}+(3-4\alpha)x)\frac{\mbox{d}}{\mbox{d}x}\right],\] and prove a Plancherel theorem. Moreover, we define the localization operators \(L_{g,\xi}\) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we give a Shapiro-type uncertainty inequality for the modified Whittaker-Stockwell transform \(\mathscr{S}_g\).

This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we derive a corresponding Plancherel’s theorem specific to the Widder–Lambert transform. This novel approach highlights an intriguing connection between these integral transforms, offering new insights into their role in harmonic analysis. Additionally, we explore a class of functions that satisfy Salem’s equivalence to the Riemann hypothesis, providing a deeper understanding of the interplay between such equivalences and integral transforms. These findings open new avenues for further research on the Riemann hypothesis within the framework of integral transforms.

Abstract This article is intended as an introduction to the distorted Fourier transform associated with a Schrödinger operator on the line or the half-line. This versatile tool has seen numerous applications in nonlinear PDE in recent years. It typically arises in the asymptotic stability analysis of topological solitons in classical field theory, such as kinks in the sine-Gordon or ϕ 4 models. The distorted Fourier transform is also a natural technique in the analysis of dispersive equations on manifolds with symmetries. Such models appear in general relativity, for example in the study of waves on a black-hole background. While microlocal methods have proven to be powerful in such applications, the more classical Weyl–Titchmarsh spectral theory and with it, the distorted Fourier transform, continue to play an essential role in the analysis of evolution PDEs. This article explain how it can be derived from Stone’s formula, which also establishes the Plancherel and inversion theorems.

The one-dimensional quaternion fractional Fourier transform (1DQFRFT) introduces a fractional-order parameter that extends traditional Fourier transform techniques, providing new insights into the analysis of quaternion-valued signals. This paper presents a rigorous theoretical foundation for the 1DQFRFT, examining essential properties such as linearity, the Plancherel theorem, conjugate symmetry, convolution, and a generalized Parseval’s theorem that collectively demonstrate the transform’s analytical power. We further explore the 1DQFRFT’s unique applications to probabilistic methods, particularly for modeling and analyzing stochastic processes within a quaternionic framework. By bridging quaternionic theory with probability, our study opens avenues for advanced applications in signal processing, communications, and applied mathematics, potentially driving significant advancements in these fields.

In this paper, we investigate the Plancherel theorem, inversion formula, and a shift operator associated with the Hartley integral transform. Building on these results, we introduce Hartley–Lipschitz type classes and derive analogues of Titchmarsh's theorems within the framework of the Hartley integral transform.

The main objective of this paper is to develop a new harmonic analysis related to a Fourier–Jacobi type operator Δ α , β m of the real line. We define and study the linear canonical Fourier–Jacobi transform F α , β m . We study some important properties, inversion formula, Plancherel theorem, Paley–Wiener theorem and Riemann–Lebsgue lemma. An application in solving a generalized heat equation is given.

We introduce the combinatorial model of J-folded alcove paths in an affine Weyl group and construct representations of affine Hecke algebras using this model. We study boundedness of these representations, and we state conjectures linking our combinatorial formulae to Kazhdan-Lusztig theory and Opdam’s Plancherel Theorem.

Clifford-wavelet transform in $L^2$-spaces is defined in space-time algebra $Cl_{(3,1)}$ of Minkowski space with orthonormal vector basis. The properties of Clifford-wavelet transform are established. Plancherel's theorem and reproducing kernel is demonstrated. The inversion formula for Clifford-wavelet transform is established. The study is supported with examples and applications from Mathematical Physics.

We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds M= SL(2,R) and M= SL(2,R)/U(1) to a finite-dimensional simple Lie group G. This construction is achieved through two equivalent ways: by means of the Plancherel Theorem and by identifying a Hilbert basis within L2(M). We analyze the existence of central extensions and identify those in duality with Hermitean operators on M. By inspecting the Clebsch–Gordan coefficients of sl(2,R), we derive the Lie brackets characterising the corresponding generalised Kac-Moody algebras. The root structure of these algebras is identified, and it is shown that an infinite number of simultaneously commuting operators can be defined. Furthermore, we briefly touch upon applications of these algebras within the realm of supergravity, particularly in scenarios where the scalar fields coordinatize the non-compact manifold SL(2,R)/U(1).

The Kontorovich-Lebedev transformation $$(KLf)(x)=\int_0^\inftyK_{i\tau}(x)f(\tau)d\tau, \;\, x \in {\mathbf R}_+$$ is consideredas an operator, which maps the weighted space $L_p(\mathbf R_+;$$\omega(\tau)d\tau), \;\, 2 \le p \le \infty$ into the Sobolevtype space $S_p^{N, \alpha}({\mathbf R}_+)$ with the finite norm$$||u||_{S_p^{N,\alpha}({\mathbf R}_+)}= \biggl( \sum_{k= 0}^N\int_0^\infty |A_x^k u|^p x^{\alpha_k p -1} dx\biggr)^{1/p} <\infty,$$where $\alpha= (\alpha_0, \alpha_1, \dots, \alpha_N), \alpha_k \in{\mathbf R}, k=0, \dots, N$, and $ A_x$ is the differentialoperator of the form$$A_x u= x^2u(x) - x\frac{d}{dx}\biggl[x\frac{du}{dx }\biggr], $$and $A_x^k$ means $k$-th iterate of $A_x, \ A_x^0u= u$. Elementary properties for the space $S_p^{N, \alpha} ({\mathbf R}_+)$ are derived. Boundedness and inversion properties for the Kontorovich-Lebedev transform are studied. In the Hilbert case ($p=2$) the isomorphism between these spaces is established for the special type of weights and Plancherel's type theorem is proved. 2000 Mathematics Subject Classification. 44A15, 46E35, 26D10

In this paper, the discrete octonion linear canonical transform (DOCLCT) is defined. According to the definition of the DOCLCT, some properties associated with the DOCLCT are explored, such as linearity, scaling, boundedness, Plancherel theorem, inversion transform and shift transform. Then, the relationship between the DOCLCT and the three-dimensional (3-D) discrete linear canonical transform (DLCT) is obtained. Moreover, based on a new convolution operator, we derive the convolution theorem of the DOCLCT. Finally, the correlation theorem of the DOCLCT is established.

In this paper, based on the biquaternion algebra, we proposed three kinds of biquaternion Fourier transforms (BiQFTs). These transforms are the extension of the complex Fourier transform. Then, the relationships between the three kinds of transforms are obtained, and it is shown that the transform can be computed by four complex Fourier transforms. Next, the inversion transforms and Plancherel theorems of the BiQFTs are proved. Moreover, the convolution theorems of the BiQFTs are studied by new convolution operators of the biquaternion. Finally, according to the convolution operator and convolution theorem associated with the right-side BiQFT, the biquaternion linear time-invariant systems are analyzed, and the biquaternion linear time-invariant systems for the right-side BiQFT is verified by the actual signal.

We generalize Beurling's theorem to the octonion Fourier transform for octonion‐valued functions. Despite challenges such as the failure of the octonion Fourier transform to change differentiation into multiplication and the absence of the octonionic Plancherel's theorem, we establish the octonionic Beurling theorem for Hermite functions on by introducing an integral condition involving the octonion Fourier transform. This extension of uncertainty principles by Hardy, Gelfand–Shilov, and Cowling–Price to the octonionic setting demonstrates that the function and its Fourier transform cannot have arbitrary Gaussian decay simultaneously.

Abstract We define and study the Stockwell transform S g \mathscr{S}_{g} associated to the Dunkl–Weinstein operator Δ k , β \Delta_{k,\beta} and prove a Plancherel theorem and an inversion formula. Next, we define a reconstruction function f Δ f_{\Delta} and prove Calderón’s reproducing inversion formula for the Dunkl–Weinstein–Stockwell transform S g \mathscr{S}_{g} . Moreover, we define the localization operators L g ⁢ ( σ ) \mathcal{L}_{g}(\sigma) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we introduce and study the extremal function F η , k ∗ := ( η ⁢ I + S g ∗ ⁢ S g ) − 1 ⁢ S g ∗ ⁢ ( k ) F^{\ast}_{\eta,\smash{k}}:=(\eta I+\mathscr{S}^{\ast}_{g}\mathscr{S}_{g})^{-1}\mathscr{S}^{\ast}_{g}(k) , and we deduce best approximate inversion formulas for the Dunkl–Weinstein–Stockwell transform S g \mathscr{S}_{g} on the Sobolev space H k , β s ⁢ ( R + d + 1 ) \mathscr{H}^{s}_{k,\beta}(\mathbb{R}_{+}^{d+1}) .

An uncertainty principle on the Lorentz spaces

The Poisson geometry of Plancherel formulas for triangular groups

Hyperbolic linear canonical transforms of quaternion signals and uncertainty