Uncertainty Principles for Fourier Multipliers
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(2020) 26:76
Uncertainty Principles for Fourier Multipliers Michael V. Northington1 Received: 30 November 2019 / Revised: 21 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The admittable Sobolev regularity is quantified for a function, w, which has a zero in the d-dimensional torus and whose reciprocal u = 1/w is a ( p, q)-multiplier. Several aspects of this problem are addressed, including zero-sets of positive Hausdorff dimension, matrix valued Fourier multipliers, and non-symmetric versions of Sobolev regularity. Additionally, we make a connection between Fourier multipliers and approximation properties of Gabor systems and shift-invariant systems. We exploit this connection and the results on Fourier multipliers to refine and extend versions of the Balian–Low uncertainty principle in these settings. Keywords Fourier multiplier · Balian–Low theorem · Uncertainty principle Mathematics Subject Classification Primary 42C15
1 Introduction Let F denote the Fourier transform on L 1 (Td ). Given 2 ≤ q ≤ ∞, a function u ∈ L 2 (Td ) is called a (2, q)-Fourier multiplier, or (2, q)-multiplier for short, if the operator Tu defined by Tu a = F(uF −1 a)
(1)
is a bounded operator from 2 (Zd ) to q (Zd ). The family of all (2, q)-multipliers q is denoted by M2 and is a Banach space when endowed with the operator norm uMq = Tu 2 (Zd )→q (Zd ) . Fourier multipliers are a classical subject in analysis 2 (see e.g. [32,33]). The study of Fourier multipliers from a p-normed space to a different q-normed space goes back to Devinatz and Hirschman [17,24] and to Hörmander [29].
Communicated by Dave Walnut.
B 1
Michael V. Northington [email protected] School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA 0123456789().: V,-vol
76
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Journal of Fourier Analysis and Applications
(2020) 26:76
A function u ∈ L ∞ (Td ) is clearly a (2, q)-multiplier for every value of q ≥ 2. In fact, it is readily checked that M22 = L ∞ (Td ). However, if u is not bounded then the situation becomes more delicate, and one may suspect that if u ‘grows rapidly’ near its ‘singularities’ then u will not be a (2, q)-multiplier, at least for certain values of q. All results in this paper are joint work with Shahaf Nitzen and Alex Powell. The goals of this paper are twofold. First, as described above, we study growth restrictions on a (2, q)-multiplier u. More precisely, we assume that u = 1/w for some function w with a zero in the d-dimensional torus, and we quantify how smooth w can be in the sense of Sobolev regularity. Our results extend in several directions including: zero sets of positive Hausdorff dimension, general ( p, q)-multipliers, non-symmetric versions of Sobolev regularity, and matrix valued Fourier multipliers. Our second goal is to develop a machinery by which one can relate such results on Fourier multipliers to problems on the uncertainty principle in time–frequency analysis. We investigate trade-offs between the time–frequency localization of the wind
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