Search

In this paper, the boundedness and compactness of the Hankel wavelet multiplier associated with the unitary representation on Lp space for 1 ≤ p ≤∞, are obtained by using the Hankel transform technique. The application of Hankel wavelet multipliers in form of the Landau–Pollak–Slepian operator is given. With the help of the Hankel wavelet multiplier, the Sobolev space associated with the Hankel transform is constructed and its various properties are studied. [ABSTRACT FROM AUTHOR]

Let n , m ≥ 1 and α > 0 . We denote by F α , m the m-analytic Bargmann–Segal–Fock space, i.e., the Hilbert space of all m-analytic functions defined on C n and square integrables with respect to the Gaussian weight exp (- α | z | 2) . We study the von Neumann algebra A of bounded linear operators acting in F α , m and commuting with all "horizontal" Weyl translations, i.e., Weyl unitary operators associated to the elements of R n . The reproducing kernel of F 1 , m was computed by Youssfi [Polyanalytic reproducing kernels in C n , Complex Anal. Synerg., 2021, 7, 28]. Multiplying the elements of F α , m by an appropriate weight, we transform this space into another reproducing kernel Hilbert space whose kernel K is invariant under horizontal translations. Using the well-known Fourier connection between Laguerre and Hermite functions, we compute the Fourier transform of K in the "horizontal direction" and decompose it into the sum of d products of Hermite functions, with d = n + m - 1 n . Finally, applying the scheme proposed by Herrera-Yañez, Maximenko, Ramos-Vazquez [Translation-invariant operators in reproducing kernel Hilbert spaces, Integr. Equ. Oper. Theory, 2022, 94, 31], we show that F α , m is isometrically isomorphic to the space of vector-functions L 2 (R n) d , and A is isometrically isomorphic to the algebra of matrix-functions L ∞ (R n) d × d . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to find the inversion formula for the covariant transform Wρk φ0. This formula is equivalent to the decomposition of the unitary representation ρk into irreducible components. We consider an eigenvalue 1+s² of the Casimir operator: dρk(C) = −4v2 (∂²u + ∂²v), where k = 0. To find the inversion formula, first we study the representations of SL2(R), ρk and ρτ, induced from the complex characters of K and N respectively. Then, we find the induced covariant transform Wρk φ0 with N-eigenvector to obtain a transform in the space L2 (SL2(R)/N). Thereafter, we compute the contravariant transform with K-eigenvector Mρτ ϕ0 : L2 (SL2(R)/K) → L2 (SL2(R)/K). [ABSTRACT FROM AUTHOR]

This is the first in a series of papers on projective positive energy representations of gauge groups. Let Ξ→M be a principal fiber bundle, and let Γc(M,Ad(Ξ)) be the group of compactly supported (local) gauge transformations. If P is a group of 'space-time symmetries' acting on Ξ→M, then a projective unitary representation of Γc(M,Ad(Ξ))⋊P is of positive energy if every 'timelike generator' p0∈p gives rise to a Hamiltonian H(p0) whose spectrum is bounded from below. Our main result shows that in the absence of fixed points for the cone of timelike generators, the projective positive energy representations of the connected component Γc(M,Ad(Ξ))0 come from 1-dimensional P-orbits. For compact M this yields a complete classification of the projective positive energy representations in terms of lowest weight representations of affine Kac-Moody algebras. For noncompact M, it yields a classification under further restrictions on the space of ground states. In the second part of this series we consider larger groups of gauge transformations, which contain also global transformations. The present results are used to localize the positive energy representations at (conformal) infinity. [ABSTRACT FROM AUTHOR]

Let G be a unipotent algebraic group defined over a p -adic field of characteristic zero. We denote by G the set of rational points of G. It is a p -adic Lie group with Lie algebra denoted by . Let π be an irreducible unitary representation of G in a Hilbert space ℋ π , f be a linear form on and be a polarization at f. We denote by χ f a character of H = exp () related to f. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors (ℋ π − ∞ ) H , χ f of  π. [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]

In this paper, we obtain explicit branching laws for all irreducible unitary representations of G=\operatorname {Spin}(N,1) when restricted to a parabolic subgroup P. The restriction turns out to be a finite direct sum of irreducible unitary representations of P. We also verify Duflo's conjecture for the branching laws of discrete series representations of G with respect to P. That is to show: in the framework of the orbit method, the branching law of a discrete series representation is determined by some geometric behavior of the moment map for the corresponding coadjoint orbit. [ABSTRACT FROM AUTHOR]

We geometrically derive the explicit form of the unitary representation of the Poincaré group for vector-valued wave functions and use it to apply speed-of-light boosts to a simple polarization basis to end up with a Hawton–Baylis photon position operator with commuting components. We give explicit formulas for other photon boost eigenmodes. We investigate the underlying affine connections on the light cone in momentum space and find that while the Pryce connection is metric semi-symmetric, the flat Hawton–Baylis connection is not semi-symmetric. Finally, we discuss the localizability of photon states on closed loops and show that photon states on the circle, both unnormalized improper states and finite-norm wave packet smeared-over washer-like regions are strictly localized not only with respect to Hawton–Baylis operators with commuting components but also with respect to the noncommutative Jauch–Piron–Amrein POV measure.

It is known that the system of integer translates of ψ ∈ L 2 (R) \ { 0 } is ℓ 2 (Z) -linearly independent precisely when the corresponding periodization function is strictly positive almost everywhere. The only known proof of this result is based on certain properties of the space of functions with uniformly bounded Fourier partial sums. We show that one of these properties does not hold for the analogue of this space in higher dimensions and square summation and, therefore, this method cannot be applied to solve the problem for integer translates of function ψ ∈ L 2 (R d) for d ⩾ 2 or ℓ 2 (Z 2 d) -linear independence of Gabor systems { T k M ℓ ψ : k , ℓ ∈ Z 2 d } for ψ ∈ L 2 (R d) , with d ⩾ 1 . [ABSTRACT FROM AUTHOR]

We give a necessary and sufficient condition for a pair of parabolic subgroups P and Q of G = G L n (R) such that the tensor product of any two unitarily induced representations from P and Q are tempered. We also give an L p -estimate of matrix coefficients of the regular representations on L 2 (G / L) when L is a Levi subgroup of G. [ABSTRACT FROM AUTHOR]

We define GG-cospectrality of two GG-gain graphs (Γ,ψ)\left(\Gamma ,\psi ) and (Γ′,ψ′)\left(\Gamma ^{\prime} ,\psi ^{\prime} ), proving that it is a switching isomorphism invariant. When GG is a finite group, we prove that GG-cospectrality is equivalent to cospectrality with respect to all unitary representations of GG. Moreover, we show that two connected gain graphs are switching equivalent if and only if the gains of their closed walks centered at an arbitrary vertex vv can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph Γ\Gamma with nn vertices and mm edges, is equal to the number of simultaneous conjugacy classes of the group Gm−n+1{G}^{m-n+1}. We provide examples of GG-cospectral switching nonisomorphic graphs and we prove that any gain graph on a cycle is determined by its GG-spectrum. Moreover, we show that when GG is a finite cyclic group, the cospectrality with respect to a faithful irreducible representation implies the cospectrality with respect to any other faithful irreducible representation, and that the same assertion is false in general.

In this paper, utilizing the theory of Watson transform and Watson convolution, we explore the Watson wavelet convolution product and its related properties. The relation between the Watson Wavelet convolution product and Watson convolution is also computed. Watson wavelet transform and its inversion formula are analyzed heuristically. Watson two-wavelet multipliers and its trace class are derived from Watson wavelet convolution product [ABSTRACT FROM AUTHOR]

In this paper, we give a pedagogical presentation of the irreducible unitary representations of ℂ 4 ⋊ S p i n (4 , ℂ) , that is, of the universal cover of the complexified Poincaré group ℂ 4 ⋊ S O (4 , ℂ). These representations were first investigated by Roffman in 1967. We provide a modern formulation of his results together with some facts from the general Wigner–Mackey theory which are relevant in this context. Moreover, we discuss different ways to realize these representations and, in the case of a non-zero "complex mass", we give a detailed construction of a more explicit realization. This explicit realization parallels and extends the one used in the classical Wigner case of ℝ 4 ⋊ S p i n 0 (1 , 3). Our analysis is motivated by the interest in the Euclidean formulation of Fermionic theories. [ABSTRACT FROM AUTHOR]

We parametrize the space of double cosets of the group G L (n , k) with respect to two subgroups T−, T+ of block strictly triangular matrices. In the appendix, we consider the quasi-regular representation of G L (n , C) in L2 on T − ∖ G L (n , C) , observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free. [ABSTRACT FROM AUTHOR]

We continue our work on understanding Howe correspondences by using theta representations from p -adic groups to compact groups. We prove some results for unitary theta representations of compact groups with respect to the induction and restriction functors. [ABSTRACT FROM AUTHOR]

Let G be a locally compact abelian group with a Haar measure, and Y be a measure space. Suppose that H is a reproducing kernel Hilbert space of functions on G × Y , such that H is naturally embedded into L 2 (G × Y) and is invariant under the translations associated with the elements of G. Under some additional technical assumptions, we study the W*-algebra V of translation-invariant bounded linear operators acting on H. First, we decompose V into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces H ^ ξ , ξ ∈ G ^ , generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of V . Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to V , i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors. [ABSTRACT FROM AUTHOR]

This paper classifies the Dirac series for complex E 7 . As applications, we verify a few conjectures raised in 2011, 2019 and 2020 for this exceptional Lie group. In particular, according to Conjecture 1.1 of Barbasch and Pandžić [Dirac cohomology and unipotent representations of complex groups, Noncommutative Geometry and Global Analysis, Contemp. Math. 546, American Mathematical Society, Providence (2011), 1–22], our classification should be helpful for understanding the unitary dual of complex E 7 . [ABSTRACT FROM AUTHOR]

This paper classifies the Dirac series for complex E 7 . As applications, we verify a few conjectures raised in 2011, 2019 and 2020 for this exceptional Lie group. In particular, according to Conjecture 1.1 of Barbasch and Pandžić [Dirac cohomology and unipotent representations of complex groups, Noncommutative Geometry and Global Analysis, Contemp. Math. 546, American Mathematical Society, Providence (2011), 1–22], our classification should be helpful for understanding the unitary dual of complex E 7 . [ABSTRACT FROM AUTHOR]

Let G be a nilpotent, connected, simply connected Lie group with Lie algebra g , and π a unitary representation of G. In this article we prove that the wave front set of π coincides with the asymptotic cone of the orbital support of π , i.e. WF (π) = AC (⋃ σ ∈ supp (π) O σ) , where O σ ⊂ i g ⁎ is the coadjoint Kirillov orbit associated to the irreducible unitary representation σ ∈ G ˆ. [ABSTRACT FROM AUTHOR]

We present a powerful theorem for proving the irreducibility of tempered unitary representations of the free group. [ABSTRACT FROM AUTHOR]

In this paper, we define the wavelet multiplier and Landau–Pollak–Slepian (L.P.S) operators on the Hilbert space L 2 (G 2 , H) , where G is a locally compact abelian topological group, and H is the quaternion algebra; Also, we will investigate some of their properties. In particular, we show that they are bounded linear operators, as well in Schatten p-class spaces, 1 ≤ p ≤ ∞ , and we determine their trace class. [ABSTRACT FROM AUTHOR]

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the simple Lie group E 6 (− 14) , which is of Hermitian symmetric type. Each FS-scattered Dirac series representation of E 6 (− 14) is realized as a composition factor of certain A q (λ) module. Along the way, we have also obtained all the fully supported irreducible unitary representations of E 6 (− 14) with integral infinitesimal characters. [ABSTRACT FROM AUTHOR]

Let F be a finite field. Consider a direct sum V of an infinite number of copies of F , consider the dual space V ⋄ , i.e., the direct product of an infinite number of copies of F. Consider the direct sum V = V ⋄ ⊕ V. The object of the paper is the group GL ‾ of continuous linear operators in V. We reduce the theory of unitary representations of GL ‾ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over F. In fact we consider a certain family Q ‾ α of subgroups in GL ‾ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to Q ‾ α , and reduce this multiplication to products of linear relations. We show that this group has type I and obtain an 'upper estimate' of the set of all irreducible unitary representations of GL ‾. [ABSTRACT FROM AUTHOR]

The aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(oid) with the help of the Moore–Aronszajn theorem. The kernel constructed from the group(oid) representation is inspired by the kernel defined in terms of the convolution of functions on a locally compact group. Several illustrative examples of reproducing kernels related with unitary representations of groupoids are discussed in detail. The paper is concluded with the brief overview of the possible applications of the proposed constructions. [ABSTRACT FROM AUTHOR]

This paper computes the Dirac index of all the weakly fair A q (λ) modules of U(p, q). We find counter-examples to a conjecture of Vogan on the unitary dual of U(p, q), which was phrased by Trapa in 2001. However, we still believe that any irreducible unitary representation of U(p, q) with non-zero Dirac cohomology must be a weakly fair A q (λ) module. [ABSTRACT FROM AUTHOR]

Admissible vectors for unitary representations of locally compact groups are the basis for group frame and covariant coherent state expansions. Main tools in the study of admissible vectors have been Plancherel and central integral decompositions, of applicability only under certain separability and semifiniteness restrictions. In this work, we present a study of admissible vectors in terms of convolution Hilbert algebras valid for arbitrary unitary representations of general locally compact groups. [ABSTRACT FROM AUTHOR]

Frames which can be generated by the action of some operators (e.g. translation, dilation, modulation, ...) on a single element $f$ in a Hilbert space, called coherent frames. In this paper, we introduce a class of continuous frames in a Hilbert space $mathcal{H}$ which is indexed by some locally compact group $G$, equipped with its left Haar measure. These frames are obtained as the orbits of a single element of Hilbert space $mathcal{H}$ under some unitary representation $pi$ of $G$ on $mathcal{H}$. It is interesting that most of important frames are coherent. We investigate canonical dual and combinations of this frames

This paper proposes a method through which a family of wavelets can be obtained. This is done by choosing each member based on a random variable. The method is preferred in situations where a single mother wavelet proves inadequate and an evolving sequence of mother wavelets is needed but a priori the next member in the sequence is uncertain. The adopted approach is distinct from the way spatiotemporal wavelets are used or even the way stochastic processes have been studied using spatiotemporal wavelets. [ABSTRACT FROM AUTHOR]

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the following simple real exceptional Lie groups: EI = E 6 (6) , EIV = E 6 (− 26) , FI = F 4 (4) , FII = F 4 (− 20). Along the way, we find an irreducible unitary representation of F 4 (4) whose Dirac index vanishes, while its Dirac cohomology is non-zero. This disproves a conjecture raised in 2015 asserting that there should be no cancellation between the even part and the odd part of the Dirac cohomology. [ABSTRACT FROM AUTHOR]

Let G be a connected complex simple Lie group, and let G ^ d {\widehat{G}^{\mathrm{d}}} be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that G ^ d {\widehat{G}^{\mathrm{d}}} consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of G ^ d {\widehat{G}^{\mathrm{d}}} come from L ^ d {\widehat{L}^{\mathrm{d}}} via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G. It follows that figuring out G ^ d {\widehat{G}^{\mathrm{d}}} requires a finite calculation in total. As an application, we report a complete description of F ^ 4 d {\widehat{F}_{4}^{\mathrm{d}}}. [ABSTRACT FROM AUTHOR]

The Cantor Set supports a Borel probability measure known as the Hutchinson measure which satisfies a well known fixed point relationship (Hutchinson in Indiana University Math J 30(5):713–747, 1981). Previously it has been shown by Jorgensen and Dutkay that the Cantor set can be extended to an inflated Cantor set, R , on a subset of the real line, which supports an extended Hutchinson measure μ ¯ (Dutkay and Jorgensen in Rev. Mat. Iberoamericana 22(1):131–180, 2006). Unitary dilation and translation operators can be defined on L 2 (R , μ ¯) which satisfy the Baumslag–Solitar group relation, and give rise to a filtration of the Hilbert space L 2 (R , μ ¯) called a multi-resolution analysis (Dutkay and Jorgensen 2006). The low pass filter function corresponding to this construction can be used to produce a measure, m, on a compact topological group called the 3-solenoid, denoted S 3 (Dutkay in Trans Am Math Soc 358(12):5271–5291, 2006). The Hilbert space L 2 (S 3 , m) also admits a unitary representation of the Baumslag–Solitar group, and there exists a generalized Fourier transform between L 2 (R , μ ¯) and L 2 (S 3 , m) (Dutkay 2006). In this paper, we build off of Jorgensen and Dutkay’s work to show that the unitary operators on L 2 (S 3 , m) mentioned above are related to each other via a family of partial isometries, which satisfy properties resembling the Cuntz relations. [ABSTRACT FROM AUTHOR]

In this article, we show that any finite gyrogroup can be represented on a space of complex-valued functions. In particular, we prove that any linear representation of a finite gyrogroup on a finite-dimensional complex inner product space is unitary and hence is completely reducible using strong connections between linear actions of groups and gyrogroups. Also, we provide an example of a unitary representation of an arbitrary finite gyrogroup, which resembles the group-theoretic left regular representation. [ABSTRACT FROM AUTHOR]