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Adv. Pure Appl. Math. 3 (2012), 113–122 DOI 10.1515/APAM.2011.015 © de Gruyter 2012 A characterisation of the Weyl transform R. Lakshmi Lavanya and S. Thangavelu Abstract. A theorem of Alesker et al. says that the Fourier transform on Rn is essentially the only transform on the space of tempered distributions which interchanges convolutions and products. In this note we obtain a similar characterisation for the Weyl transform. Keywords. Schwartz class, tempered distributions, Weyl transform, Fourier–Weyl transform, noncommutative derivations. 2010 Mathematics

Mont e Carlo Methodsand Appl, Vol.2, No. 4, pp. 271-293 (1996) © VSP1996 Riesz-Raikov sums and Weyl transform Katusi Fukuyama Department of Mathematics, Kobe University, Rokko, Kobe, 657 Japan e-mail: fukuyama@math.s.kobe-u.ac.jp Abstract — We investigate the dependence of stationary sequence which is obtained from the Riesz-Raikov sum by applying the Weyl transform. One of the results solves Sugita's conjecture on quasi - Monte Carlo methods. 1. Introduction The Rademacher functions {r,·} can be regarded as i.i.d. on the Lebesgue probability space (Ω := [0 ,1

MONTE CARLO METHODS AND APPLICATIONS (ISSN 0929-9629) Vol. 2, No. 4, pp. 255-346 1996 CONTENTS A probabilistic result on the discrepancy of a hybrid-Monte Carlo sequence and applications G. Ökten 255 Riesz-Raikov sums and Weyl transform K. Fukvyama 271 On the use of low discrepancy sequences in Monte Carlo methods B. Tuffin 295 Variation of product function and numerical solutions of some partial differential equations by low-discrepancy sequences Yi-JunXiao 321 On Hamming weight test and sojourn time test of /w-sequences K. Takashima 331

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state solutions to the conserved Kuramoto–Sivashinsky equation 59 M. BOUSLIMI, K. KEFI, F.-D. PREDA Variational analysis for an indefinite quasilinear problem with variable exponent 67 J.-R. PYCKE A probabilistic counterpart of the Askey scheme for continuous polynomials 85 R. LAKSHMI LAVANYA, S. THANGAVELU A characterisation of the Weyl transform 113 Offenlegung der Inhaber und Beteiligungsverhältnisse gem. § 7a Abs. 1 Ziff. 1, Abs. 2 Ziff. 3 des Berliner Pressegesetzes: Gisela Cram, Rentnerin, Berlin; Dr. Annette Lubasch, Ärztin, Berlin; Elsbeth Cram, Pensionärin

, McGraw-Hill, Inc, 1964. [35] Shale D., Linear symmetries of free boson fields, Trans. Amer. Math. Soc., 1962, 340, 309–321. [36] Thomas T., Character of the Weil representation, 2008, 77(2), 221–239. [37] Thomas T., The Weil representation, the Weyl transform, and transfer factor, 2009. [38] Von Neumann J., Mathematical fundations of quantum mechanics, translated by Robert T. Beyer, Princeton University Press, 1955. [39] Waldspurger J.-L., Démonstration d’une conjecture de dualité de Howe dans le cas p-adique, p ≠ 2, Festschrift in honor of I. I. Piatetski

convolution, 26 twisted product, 103 uncertainty principle, 27 van Hove’s theorem, 199 wave front set, 119, 154 —, analytic, 164 wave packet transform, 143, 148 Weil-Brezin transform, 71 Weyl correspondence, 79 Weyl symbol, 80 Weyl transform, 24 Wick ordered operator, 138 Wick symbol, 138, 139 Wigner distribution, 57 Wigner transform, 56 Zak transform, 71 78 D69 085289 9780691085289

benchmark test for the comparisons between the Schrödinger and Wigner equations. We nally report the results of our simulations and discuss them. 2 Quantum dynamics in phase-space The Wigner formulation of quantum mechanics [12] oers a description of the electron state in terms of a phase-space function fw(x, k, t), where x is the position and k is the wave number (ℏk momentum) vari- able. The pure state Wigner function is related to the solution of the Schrödinger equation ×(x, t) via the Wigner–Weyl transform: fw(x, k, t) = 1 iℏ2ð ∫ dxe−ikx ×(x + x 2 , t)×∗(x − x 2 , t

theorem (compare with Theorem 2.6 of Strichartz [7]). 2π Theorem 2.1. Assume that feL2(#„red) and J/(z, i)dt = 0. Then we have o oo ±00 f. To prove the theorem we need to recall the following result. Let φ£ ~ l(z) be defined by (2.4) <pZ~l(z) = L%~ and let/x φζ"1 be the twisted convolution _ n — l / \ f / Y \ n—is \ —*mz' Cn 54 Then one has (2.6) Tkangavelu, Resiriction theoremsfor the Heisenberg group ll/lli This follows from the Plancherei theorem for the Weyl transform (see [5]). i Let KJ1^) = <t>k~l(j~2z) and/'(z) be defined by 2ir (2.7) /'(z)- J **/(*,/)A. 0 Then a

functions were studied by Bownik in [ 4 ]. The study of shift-invariant spaces and frames has been extended to locally compact abelian groups in [ 5 , 14 ] and non-abelian compact groups in [ 17 ]. Radha and Adhikari [ 16 ] introduced twisted shift-invariant spaces in L 2 ⁢ ( ℝ 2 ⁢ n ) {L^{2}(\mathbb{R}^{2n})} and studied characterizations of orthonormal systems, Bessel sequences, frames and Riesz bases of twisted translates in terms of the kernel of the Weyl transform. The twisted translation and twisted shift-invariant space are defined as follows. Definition 1

CHAPTER 1 HERMITE, SPECIAL HERMITE AND LAGUERRE FUNCTIONS In this chapter we define Hermite, Laguerre and special Hermite func- tions and prove some of their properties which are needed in studying the expansions in terms of them. In order to derive the special Hermite functions we recall briefly some results from the representation theory of the Heisenberg group. We also define and prove some properties of Weyl transforms which we need in the study of special Hermite expansions. Some important asymptotic properties and norm estimates are collected in Section 1