Zeros of the Wigner distribution and the short-time Fourier transform
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Zeros of the Wigner distribution and the short-time Fourier transform Karlheinz Gröchenig1
· Philippe Jaming2 · Eugenia Malinnikova3,4
Received: 12 December 2018 / Accepted: 5 November 2019 © The Author(s) 2019
Abstract We study the question under which conditions the zero set of a (cross-) Wigner distribution W ( f , g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson’s theorem for the positivity of the Wigner distribution and to Hardy’s uncertainty principle. We then construct a class of step functions S so that the Wigner distribution W ( f , 1(0,1) ) always possesses a zero f ∈ S ∩ L p when p < ∞, but may be zerofree for f ∈ S ∩ L ∞ . The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis. Keywords Wigner distribution · Short-time Fourier transform · Hudson’s theorem · Poly-analytic function · Convexity · Hurwitz polynomial · Totally positive function Mathematics Subject Classification 42A38 · 42B10 · 81S30
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Karlheinz Gröchenig [email protected] Philippe Jaming [email protected] Eugenia Malinnikova [email protected]
1
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
2
Institut de Mathématiques de Bordeaux UMR 5251, Université de Bordeaux, Cours de la Libération, 33405 Talence Cedex, France
3
Department of Mathematical Sciences, Norwegian University of Science and Technology, Alfred Getz vei 1, Trondheim, Norway
4
Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, USA
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K. Gröchenig et al.
1 Introduction The aim of this paper is to study the zero set of the Wigner distribution of two functions f , g ∈ L 2 (R), W ( f , g)(z) =
Rd
f (x + 2t )g(x ¯ − 2t )e−2πiξ,t dt,
z = (x, ξ ) ∈ R2d .
(1)
More precisely, we are investigating whether this zero set can be empty. Results here directly extend to other phase-space representations to which the Wigner transform is closely related. These include the ambiguity function and the short-time Fourier transform Vg f (z) = f , π(z)g L 2 (Rd ) , π(z)g(t) = e2iπ ξ,t g(t − x) for z = (x, ξ ) ∈ R2d (see Eq. 2). The zero set of the short-time Fourier transform is important in the study of the generalized Berezin quantization and the injectivity of a general Berezin transform. The thesis of D. Bayer [5], partially published in [6], contains the following result (under a mild condition on f , g ∈ L 2 (Rd )): If Vg f (z) = 0 for all z ∈ R2d , then the mapping T → BT , with z → BT (z) = T π(z)g, π(z) f , is one-to-one on the space of bounded operators on L 2 (Rd ). The function (z, w) → T π(z)g, π(w) f may be interpreted a
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