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We prove in this paper a generalization of Hardy's theorem for Gabor transform in the setup of the semidirect product R n ⋊ K , where K is a compact subgroup of automorphisms of R n . We also solve the sharpness problem and thus obtain a complete analogue of Hardy's theorem for Gabor transform. The representation theory and Plancherel formula are fundamental tools in the proof of our results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Titchmarsh theorem for Laguerre hypergroup K = [ 0 , + ∞ [ × R , via moduli of continuity of higher orders. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for functions defined on the space L p (R , d μ) , 1 < p ≤ 2 , providing the weighted integrability of their Cherednik-Opdam transforms. These results generalize a famous Titchmarsh’s theorem and Younis’ theorem for functions from Lipschitz classes. [ABSTRACT FROM AUTHOR]

In this paper, we obtain sufficient conditions for the weighted integrability of Jacobi–Dunkl transforms in terms of the moduli of smoothness connected with Jacobi–Dunkl translation operators. These results generalize a famous Titchmarsh's theorem and Younis' theorem for functions from Lipschitz classes. [ABSTRACT FROM AUTHOR]

The problem of weighted integrability of the Laguerre–Bessel transform in terms of the moduli of smoothness related to generalized translations is considered. Sufficient conditions are given to solve this problem. These results generalize a famous Titchmarsh’s theorem, due to [L. Rakhimi and R. Daher, An analog of Titchmarsh’s theorem for the Laguerre–Bessel transform, Arab. J. Math. Sci. 2023, 10.1108/AJMS-04-2022-0101] in the Laguerre–Bessel hypergroup. Also some results connected with the integrability of Laguerre–Bessel transforms are given. [ABSTRACT FROM AUTHOR]

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform. First, we prove that the hyperbolic Bloch transform is injective and "asymptotically unitary" already in the simplest case—that is, when the Hilbert space is the regular representation of the Fuchsian group, Γ . Second, when Γ ⊂ PSU (1 , 1) acts isometrically on the hyperbolic plane H and the Hilbert space is L 2 H , we define a modified, geometric Bloch transform that sends wave functions to sections of irreducible, flat, Hermitian vector bundles over Σ = H / Γ and transforms the hyperbolic Laplacian into the covariant one. [ABSTRACT FROM AUTHOR]

We study structural properties and the harmonic analysis of discrete subgroups of the Euclidean group. In particular, we 1. obtain an efficient description of their dual space, 2. develop Fourier analysis methods for periodic mappings on them, and 3. prove a Schur-Zassenhaus type splitting result. [ABSTRACT FROM AUTHOR]

Pitt’s inequality for exponential solvable Lie groups with non-trivial center, connected nilpotent Lie groups with non-compact center, Heisenberg motion group and diamond Lie groups has been proved. These inequalities have been used to establish logarithmic uncertainty inequality and Heisenberg uncertainty inequality for the above classes of groups. [ABSTRACT FROM AUTHOR]

In this note, we introduce Sobolev spaces over (not necessarily abelian) compact groups and study their properties. In particular, we focus on Sobolev embedding and compactness theorems. As an application, we prove the existence of solutions for the generalized Bosonic equations posed over compact groups. [ABSTRACT FROM AUTHOR]

In this paper, we describe the cyclic 3-nacci sequence and the cyclic generalized order-3 Pell sequence in the non-abelian groups and then we investigate them in non-abelian finite groups in detail. Furthermore, we give the lengths of the periods of the cyclic 3-nacci sequence and the cyclic generalized order-3 Pell sequence in the polyhedral group (n , 2 , 2) with respect to the generating triple (a , b , c) as applications of the results produced. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary and sufficient conditions in terms of Fourier–Bessel coefficients of a function f, to ensure that f belongs either to one of the generalized Lipschitz classes H α m and h α m for α > - 1 / 2 . Also, a condition for generalized Bessel differentiability of a function on interval [0, 1] is proved. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary and sufficient conditions in terms of Fourier–Jacobi coefficients of a function f, to ensure that f belongs either to one of the generalized Lipschitz classes H α , β m and h α , β m for α ≥ β ≥ − 1 / 2 and α ≠ − 1 / 2. Also a condition for generalized Jacobi differentiability of a function on interval [ 0 , π ] is proved. [ABSTRACT FROM AUTHOR]

Wigner-Ville transform or Wigner-Ville distribution (WVD) associated with quaternion offset linear canonical transform (QOLCT) was proposed by Bhat and Dar. This work is devoted to the development of the theory proposed by them, which is an emerging tool in the scenario of signal processing. The main contribution of this work is to introduce WVD and ambiguity function (AF) associated with the QOLCT (WVD-QOLCT/AF-QOLCT). First, the definition of the WVD-QOLCT is proposed, and then several important properties such as dilation, nonlinearity, and boundedness are derived. Second, we derived the AF for the proposed transform. A bunch of important properties, including the reconstruction formula associated with the AF, are studied.

In this paper, we give a characterization of the kernel of τ in a trivolutive Banach algebra (A , τ) . The main results (under a certain condition) are: (a) The closed ideal I = ker τ of a unital Banach algebra A is principal. (b) The closed ideal I = ker τ has a bounded approximate identity if A has a bounded approximate identity. Also we give some concrete example which indicates that the necessary condition must occur. Moreover, we relate trivolutions on the algebra A × θ B with θ -Lau product to the corresponding ones on A and B . [ABSTRACT FROM AUTHOR]

This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces H s (R) is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in H 1 2 (R) with mass strictly less than 4 π or general initial conditions in the weighted Sobolev space H 2 , 2 (R) . In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in H 1 2 (R) and that furthermore the H 1 2 norm of the solutions remains globally bounded in time. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation. [ABSTRACT FROM AUTHOR]

The set X of k-subsets of an n-set has a natural graph structure where two k-subsets are neighbors if and only if the size of their intersection is k - 1 . This is known as the Johnson graph. The symmetric group S n acts on the space of complex functions on X and this space has a multiplicity-free decomposition as sum of irreducible representations of S n , so it has a well-defined Gelfand–Tsetlin basis up to scalars. The Fourier transform on the Johnson graph is defined as the change of basis matrix from the delta function basis to the Gelfand–Tsetlin basis. The direct application of this matrix to a generic vector requires n k 2 arithmetic operations. We show that, in analogy with the classical Fast Fourier Transform on the discrete circle, this matrix can be factorized as a product of n - 1 orthogonal matrices, each one with at most two nonzero elements in each column. The factorization is based on the construction of n - 1 intermediate bases which are parametrized via the Robinson–Schensted insertion algorithm. This factorization shows that the number of arithmetic operations required to apply this matrix to a generic vector is bounded above by 2 (n - 1) n k . We give an algorithm that constructs all these factors using at most 289 (n - 1) n k arithmetic operations. The coefficients of these matrices are rational numbers and the construction does not depend on numerical methods. Instead, they are obtained by solving small linear systems with integer coefficients derived from the Jucys–Murphy operators. In particular we avoid the use of square roots. As a consequence, we show that the problem of computing all the weights of the isotypic components of a given function can be solved in O (n n k ) operations, improving the previous bound O (k 2 n k ) when k asymptotically dominates n . The same improvement is achieved for the problem of computing the isotypic projection onto a single component. [ABSTRACT FROM AUTHOR]

The quaternion offset linear canonical transform (QOLCT) has gained much popularity in recent years because of its applications in many areas, including image and signal processing. At the same time, the applications of Wigner–Ville distribution (WVD) in signal analysis and image processing cannot be excluded. In this paper, we investigate the Wigner–Ville distribution associated with quaternion offset linear canonical transform (WVD-QOLCT). Firstly, we propose the definition of the WVD-QOLCT, and then, several important properties of newly defined WVD-QOLCT, such as reconstruction formula, orthogonality relation, are derived. Secondly, a novel canonical convolution operator and a related correlation operator for WVD-QOLCT are proposed. Based on the proposed operators, the corresponding generalized convolution and correlation theorems are studied. Moreover on the application part, detection of the linear frequency modulated signals is established in detail by constructing an example. [ABSTRACT FROM AUTHOR]

In a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the k-cycle shuffle, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008). [ABSTRACT FROM AUTHOR]

In this paper, we extend the classical Titchmarsh theorem on the image under the discrete Fourier–Laplace transform of a set of functions satisfying a generalized Lipschitz and Dini–Lipschitz condition in the space S (p , q) (σ m - 1) , m ≥ 3 . The given statements generalize the results of the author’s work published in El Ouadih and Daher (Integral Transforms Spec Funct 31(12):1010–1019, 2020). [ABSTRACT FROM AUTHOR]

Our aim in this paper is to prove an analogue of the classical Titchmarsh theorem on the image under the discrete Fourier–Laplace transform of a set of functions satisfying a generalized Lipschitz condition in the space L 2 on the sphere. [ABSTRACT FROM AUTHOR]

The purpose of this work is to prove an analog of the classical Titchmarsh's theorem (Introduction to the theory of Fourier integrals, Oxford University Press, Oxford, 1937, Theorem 84) and Younis's Theorem (Fourier transform of Lipschitz functions on compact groups, Ph.D. Thesis, McMaster University, Hamilton, Ontario, Canada, 1974, Theorem 2.6) on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition in the weighted spaces L p ([ 0 , π ]) , 1 < p ≤ 2 . For this purpose, we use a generalized translation operator which was defined by Flensted-Jensen and Koornwinder in (The convolution structure for Jacobi function expansions, Ark. Mat., 1973) [ABSTRACT FROM AUTHOR]

Let M be a semi‐finite von Neumann algebra with normal state space S(M). For any ϕ∈S(M), let Mϕ:={x∈M:xϕ=ϕx} be the centralizer of ϕ with center Z(Mϕ). We show that for ϕ,ψ∈S(M), the following are equivalent. ϕ=ψ.Z(Mψ)⊆Z(Mϕ) and ϕ|Z(Mϕ)=ψ|Z(Mϕ).ϕ,ψ have the same distances to all the closed faces of S(M). As an application, we give an alternative proof of the fact that metric preserving surjections between normal state spaces of semi‐finite von Neumann algebras are induced by Jordan ∗‐isomorphisms between the underlying algebras. We then use it to verify some facts concerning F‐algebras and Fourier algebras of locally compact quantum groups. [ABSTRACT FROM AUTHOR]

In this paper, we investigate key assumptions on some Banach algebras to have quasi-weak normal structure (resp. quasi-weak ⋆ normal structure) and to prove in particular that fixed point property for Kannan mappings is satisfied on weakly compact convex subsets in this setting. We establish some conditions on a locally compact group G for which the Fourier and Fourier–Stieltjes–Banach algebras A(G) and B(G) have quasi-normal structure. In addition, we give some examples of Banach spaces X such that L (X) (the Banach algebra of bounded linear operators on X) and some of its closed two-sided ideals associated with Fredholm perturbations have fixed point properties. Finally, we give some comments and interesting questions related to this area. [ABSTRACT FROM AUTHOR]

This paper introduces a systematic study for analytic aspects of discrete spectra methods for functions supported on some compact domains of S E (2) , according to Sturm–Liouville theory. We then apply these discrete spectra methods to approximate convolution of functions supported on these compact domains. We shall also investigate different aspects of the presented theory in the cases of zero-value boundary condition and derivative boundary condition. The paper is concluded by some Plancherel formulas associated to Sturm–Liouville theory and special cases of boundary conditions. [ABSTRACT FROM AUTHOR]