The Harmonic Analysis Seminar takes place this semester on Tuesdays from 11:30 - 13:00 (CEST), on site in SR10 (OMP1, second floor) and it will not be streamed online.

This semester, the goal is to study the topic of systems of exponential functions and their spanning properties (detailed description below). The seminar is offered as a graduate class within the Vienna School of Mathematics (VSM) and only basic knowledge of Fourier analysis is assumed.

Organizers: José Luis Romero and Jordy van Velthoven

The goal of this seminar is to discuss important notions and results on function systems consisting of complex exponentials. This allows the participants to build up a working knowledge for research in Fourier analysis.

A classical result from Fourier analysis asserts that the system of exponentials with integer frequencies constitutes an orthonormal basis for the space of square-integrable functions on the unit interval. This property generally fails already after small perturbations of the frequencies and leaves a system of exponentials forming a nonorthonormal basis. The study of exponential systems on

other domains than (unions of) intervals (”disconnected spectra”) necessitates the study of weaker notions than bases, namely that of frames (”overcomplete bases”) and Riesz sequences (”undercomplete bases”). The significance of a frame is that it still guarantees reconstruction of a function from its values on a certain set (sampling), whereas the Riesz property allows to prescribe the values of a function on a given set (interpolation). The seminar will address the basic theory of frames and Riesz sequences of exponentials.

The required background consists only of basic knowledge of Fourier analysis and Hilbert spaces.

Reading list:
- Bownik, M., Londner, I., On syndetic Riesz sequences. Isr. J. Math. 233, No. 1, 113-131 (2019).
- Kozma, G., Nitzan, S., Olevskiˇı, A., A set with no Riesz basis of exponentials. Rev. Mat. Iberoam. 39, No. 6, 2007-2016 (2023).
- Kozma, G., Nitzan, S., Combining Riesz bases. Invent. Math. 199, No. 1, 267-285 (2015).
- Landau, H. J., Necessary density conditions for sampling an interpolation of certain entire functions. Acta Math. 117, 37-52 (1967).

- Matei, B; Meyer, Y, A variant of compressed sensing. Rev. Mat. Iberoam. 25, No. 2, 669-692 (2009).
- Nitzan, S., Olevskii, A., Revisiting Landau’s density theorems for Paley-Wiener spaces. C. R., Math., Acad. Sci. Paris 350, No. 9-10, 509-512 (2012).
- A. Olevskii, A. Ulanovskii, Functions with Disconnected Spectrum, University Lecture Series 65. Providence, RI: American Mathematical Society (AMS). x, 138 p. (2016).