Modern aspects of Fourier analysis
SEMINARS
This seminar series is part of the
Intensive Research Programme on Modern Trends in Fourier Analysis
DESCRIPTION
During the IRP there will be two seminars per week: Tuesdays at 15h and Fridays at 12h. They will be held at C1/028 at CRM.
Tuesday 6th May | Some new insights into weighted Sobolev inequalities
Andrea Olivo del Valle
BCAM
ABSTRACT
It is well known that the celebrated Gagliardo estimate can be viewed as an extension of the classical isoperimetric inequality, although the best constant was not obtained by Gagliardo. In this talk, we will explore some generalizations of this result involving weights and discuss how it can be extended beyond smooth domains. We will also discuss a certain type of weighted Sobolev inequalities, inspired by a work of David-Semmes.
Friday 9th May | Multilinear singular integral operators with rough kernels
Lenka Slavikova
Charles University Prague
ABSTRACT
In this talk, I will review recent results involving boundedness properties of multilinear singular integral operators associated with rough homogeneous kernels. Emphasis will be put on the case when the restriction of the underlying singular integral kernel to the unit sphere belongs to the space L^q, where q>1. I will also discuss the use of less standard function spaces in this connection. The talk is based on joint projects with G. Dosidis, L. Grafakos, D. He, P. Honzík, S. Lappas and B. Park
Tuesday 13th May | Pauli pairs and Fourier Uniqueness
João Pedro Ramos
EPFL, King’s College London
ABSTRACT
Motivated by the classical Pauli problem of determining a function $f$ (up to global phase) from the magnitudes $|f|,|\widehat{f}|$, we shall investigate a related question: given the values of $|f|$ and $|\widehat{f}|$ on discrete sequences, when is it possible to recover the values of $|f|,|\widehat{f}|$ on the whole real line?
By employing techniques recently developed in the context of Fourier Uniqueness Pairs, we shall see several necessary and sufficient conditions for the property above to hold. Time-permitting, we shall also see how such results generalize to higher dimensions and have direct consequences to problems of recovery-type for partial differential equations.
This talk is based on joint work with Mateus Sousa
Friday 16th May | Fourier uniqueness and interpolation for dense but irregular sets
Alexei Kulikov
Tel Aviv University
ABSTRACT
A pair of sets A, B is called a Fourier uniqueness pair for the function space X if there are no non-zero functions f in X such that f is zero on A and its Fourier transform is zero on B. Many results in Fourier analysis can be stated in this form, for example the famous Benedicks uncertainty principle says that if the complements of A and B have finite measure then they form a Fourier uniqueness pair for L^2(R).
In all classical examples of Fourier uniqueness pairs at least one of the sets had positive measure. In 2017 Danylo Radchenko and Marina Viazovska constructed the first example of a discrete Fourier uniqueness pair — they showed that the only Schwartz function such that it and its Fourier transform vanishes at all positive and negative square roots of all non-negative integers and at one more extra point is a zero function.
In a joint work with Fedor Nazarov and Mikhail Sodin we showed that if we shrink Radchenko–Viazovska sets by a constant factor strictly less than 1 we still end up with a Fourier uniqueness pair. Moreover, unlike their proof which required some number-theoretic input, our argument was purely analytic, the only assumption that we needed was a uniform bound on the differences of the consecutive elements of our sets.
In this talk I will tell about recent developments of this theme, in which we construct new Fourier uniqueness pairs that have even less structure: we can replace the assumption on the differences of the consecutive elements with its averaged version in the spirit of the uniform lower density. Curiously, the more irregular the sets get, the more regularity we need to demand from the function space for the result to hold.
Friday 23rd May | Sampling theorems in shift-invariant spaces
Ilya Zlotnikov
NTNU
ABSTRACT
We will discuss two sampling theorems in the shift-invariant space generated by the bivariate Gaussian function.
First, we consider the mobile sampling problem for a collection of parallel lines
Λ := {(x, y) ∈ R^2: (x, y) · ⃗v ∈ Γ}, Γ ⊂ R is separated.
We give a necessary and sufficient condition for Λ to form a mobile sampling trajectory for V^2g(R^2) in terms of the lower uniform density of the set Γ and the direction vector v. Unlike the standard Paley–Wiener setting (see, e.g., [2]), the results are completely different for lines with rational orirrational slopes.
Second, for the space V^2g(R^2), we solve the classical sampling problems (i.e., we give necessary andsufficient conditions for a stable recovery of any signal from separated samples) for a large class of lattices spread along parallel lines with rational slope.
Finally, using the relation between sampling in shift-invariant spaces and Gabor systems generated by bivariate Gaussian, we establish new examples of Gabor frames with non-complex lattices having a volume close to critical.
References
[1] José Luis Romero, Alexander Ulanovskii, Ilya Zlotnikov, Sampling in the shift-invariant space generated by the bivariate Gaussian function, Journal of Functional Analysis, Volume 287, Issue 9, 110600, (2024) DOI: 10.1016/j.jfa.2024.110600
[2] Alexander Rashkovskii, Alexander Ulanovskii, Ilya Zlotnikov, On 2-dimensional mobile sampling, Applied and Computational Harmonic Analysis, Volume 62, 1-23, (2023) DOI: 10.1016/j.acha.2022.08.001.