Fourier Interpolation and Time-Frequency Localization

Aleksei Kulikov

doi:10.1007/s00041-021-09861-y

Abstract

We prove that under very mild conditions for any interpolation formula f(x) = sum _{lambda in Lambda } f(lambda )a_lambda (x) + sum _{mu in M} {hat{f}}(mu )b_{mu }(x) we have a lower bound for the counting functions n_Lambda (R_1) + n_{M}(R_2) ge 4R_1R_2 - Clog ^{2}(4R_1R_2) which very closely matches recent interpolation formulas of Radchenko and Viazovska and of Bondarenko, Radchenko and Seip.

Highlights

In the recent breakthrough paper [7] Radchenko and Viazovska showed that any Schwartz function can be effectiv√ely reconstructed Fourier transform at the points ± n, n ∈ Z≥0 and from the two more values values of it and its f (0), f (0).If w=e{±co√nsni}detarktehsethceoufonrtimngn function (R) =n (R) = 1 + 2[R2], | we∩ [−R, R]|, which in the case see that it satisfies the inequality n (W ) + n (T ) ≥ 4W T − O(1) for all W, T
We observe that this bound perfectly matches the famous 4W T Theorem of Slepian [10] which says that the space of functions which are supported on [−T, T ] and such that their Fourier transforms are Communicated by Hans G
Journal of Fourier Analysis and Applications (2021) 27:58 essentially supported on [−W, W ] has approximate dimension 4W T .1. We prove that this is not a coincidence and that a similar inequality holds for all such interpolation formulas with a very small error term

Summary

Introduction

In the recent breakthrough paper [7] Radchenko and Viazovska showed that any Schwartz function can be effectiv√ely reconstructed Fourier transform at the points ± n, n ∈ Z≥0 and from the two more values values of it and its f (0), f (0). ∩ [−R, R]|, which in the case see that it satisfies the inequality n (W ) + n (T ) ≥ 4W T − O(1) for all W , T. We observe that this bound perfectly matches the famous 4W T Theorem of Slepian [10] which says that the space of functions which are supported on [−T , T ] and such that their Fourier transforms are Communicated by Hans G.

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Prolate Spheroidal Wave Functions

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