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.

First reading list:
- Avdonin, S.A. On the question of Riesz bases of exponential functions in L^2, Vestnik Leningrad Univ. Math. 7 (1979)
- Bownik, M., Londner, I., On syndetic Riesz sequences. Isr. J. Math. 233, No. 1, 113-131 (2019). Link
- Iosevich, A. Fuglede conjecture for lattices. Link
- Kozma, G., Nitzan, S., Olevskii, A., A set with no Riesz basis of exponentials. Rev. Mat. Iberoam. 39, No. 6, 2007-2016 (2023). Link
- Kozma, G., Nitzan, S., Combining Riesz bases. Invent. Math. 199, No. 1, 267-285 (2015). Link
- Łaba, I. Fuglede’s conjecture for a union of two intervals. Proc. Am. Math. Soc. 129, No. 10, 2965-2972 (2001). Link
- Landau, H. J., Necessary density conditions for sampling an interpolation of certain entire functions. Acta Math. 117, 37-52 (1967). Link
- Matei, B; Meyer, Y, A variant of compressed sensing. Rev. Mat. Iberoam. 25, No. 2, 669-692 (2009). Link
- Olevskii, A., Ulanovskii, A., Functions with Disconnected Spectrum, University Lecture Series 65. Providence, RI: American Mathematical Society (AMS).  138 p. (2016).
- Young, R., An introduction to nonharmonic Fourier series. Revised edition. Orlando, FL: Academic Press. 234 p. (2001).

Second reading list:
- Iosevich, A., Pedersen, S. How large are the spectral gaps? Pac. J. Math., No. 2, 307-314 (2000). Link
- Kolountzakis, M. Multiple lattice tiles and Riesz bases of exponentials. Proc. Am. Math. Soc. 143, No. 2, 741-747 (2015). Link
- Kolountzakis, M., Lev, N., On non-periodic tilings of the real line by a function. Int. Math. Res. Not. IMRN, No. 15, 4588–4601 (2016). Link
- Komornik, V., Explicit multidimensional Ingham-Beurling type estimates. Link
- Landau, H. A sparse regular sequence of exponentials closed on large sets. Bull. Am. Math. Soc. 70, 566-569 (1964). Link
- 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). Link