82 research outputs found

    The Zak transform and some counterexamples in time-frequency analysis

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    Counterexamples to the B-spline conjecture for Gabor frames

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    The frame set conjecture for B-splines BnB_n, n≥2n \ge 2, states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form ab=rab=r, where rr is a rational number smaller than one and aa and bb denote the sampling and modulation rates, respectively, has infinitely many pieces, located around b=2,3,…b=2,3,\dots, \emph{not} belonging to the frame set of the nnth order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines BnB_n, n≥2n \ge 2.Comment: Version 2: Lem. 5, Prop. 6, and Thm. 7 added, Version 3: Thm. 8 change

    On the non-frame property of Gabor systems with Hermite generators and the frame set conjecture

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    The frame set conjecture for Hermite functions formulated in [Gr\"ochenig, J. Fourier Anal. Appl., 20(4):865-895, 2014] states that the Gabor frame set for these generators is the largest possible, that is, the time-frequency shifts of the Hermite functions associated with sampling rates α\alpha and modulation rates β\beta that avoid all known obstructions lead to Gabor frames for L2(R)L^{2}(\mathbb{R}). By results in [Seip and Wallst\'en, J. Reine Angew. Math., 429:107-113, 1992] and [Lemvig, Monatsh. Math., 182(4):899-912, 2017], it is known that the conjecture is true for the Gaussian, the 00th order Hermite functions, and false for Hermite functions of order 2,3,6,7,10,11,…2,3,6,7,10,11,\dots, respectively. In this paper we disprove the remaining cases except for the 11st order Hermite function

    On some Hermite series identities and their applications to Gabor analysis

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    We prove some infinite series identities for the Hermite functions. From these identities we disprove the Gabor frame set conjecture for Hermite functions of order 4m+24m+2 and 4m+34m+3 for m∈{0}∪Nm \in \{0\} \cup \mathbb{N}. The results hold not only for Hermite functions, but for two large classes of eigenfunctions of the Fourier transform associated with the eigenvalues −1-1 and ii, and the results indicate that the Gabor frame set of all such functions must have a rather complicated structure

    A Short Note on the Frame Set of Odd Functions

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    In this work we derive a simple argument which shows that Gabor systems consisting of odd functions of dd variables and symplectic lattices of density 2d2^d cannot constitute a Gabor frame. In the 1--dimensional, separable case, this is a special case of a result proved by Lyubarskii and Nes, however, we use a different approach in this work exploiting the algebraic relation between the ambiguity function and the Wigner distribution as well as their relation given by the (symplectic) Fourier transform. Also, we do not need the assumption that the lattice is separable and, hence, new restrictions are added to the full frame set of odd functions.Comment: accepted: Bulletin of the Australian Mathematical Society; 12 pages; Version 3 makes use of symmetric time-frequency shifts. In this case the appearing phase factors are easier to handle. Also, the main result is extended to higher dimensions. [In Version 2 a mistake in the assumptions was corrected. The windows should be chosen from Feichtinger's algebra rather than from the Hilbert space L2.

    Spectral asymptotics of periodic elliptic operators

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    We demonstrate that the structure of complex second-order strongly elliptic operators HH on Rd{\bf R}^d with coefficients invariant under translation by Zd{\bf Z}^d can be analyzed through decomposition in terms of versions HzH_z, z∈Tdz\in{\bf T}^d, of HH with zz-periodic boundary conditions acting on L2(Id)L_2({\bf I}^d) where I=[0,1>{\bf I}=[0,1>. If the semigroup SS generated by HH has a H\"older continuous integral kernel satisfying Gaussian bounds then the semigroups SzS^z generated by the HzH_z have kernels with similar properties and z↦Szz\mapsto S^z extends to a function on Cd∖{0}{\bf C}^d\setminus\{0\} which is analytic with respect to the trace norm. The sequence of semigroups S(m),zS^{(m),z} obtained by rescaling the coefficients of HzH_z by c(x)→c(mx)c(x)\to c(mx) converges in trace norm to the semigroup S^z\hat{S}^z generated by the homogenization H^z\hat{H}_z of HzH_z. These convergence properties allow asymptotic analysis of the spectrum of HH.Comment: 27 pages, LaTeX article styl

    Balian-Low Type Results for Gabor Schauder Bases

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