The Harmonic Analysis Seminar takes place this semester on Tuesdays from 11:30 - 13:00 (CEST), on site in SR9 in Kolingasse, and it will not be streamed online.

This semester, the goal is to study the topic of norms estimates of (non)harmonic Fourier series (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 various results on norms of linear combinations of complex exponentials. This allows the participants to build up a working knowledge for research in Fourier analysis.  

Parseval’s identity from Fourier analysis asserts that the 2-norm of a linear combination of complex exponentials with integer frequencies (harmonic Fourier series) is equal to the 2-norm of the coefficients. The first aim of the seminar is to obtain similar inequalities for complex exponentials with possibly noninteger frequencies (nonharmonic Fourier series), namely inequalities relating the 2-norm of a nonharmonic Fourier series and its coefficients. The central result for such nonharmonic Fourier series is Ingham’s theorem.  After studying 2-norms of nonharmonic Fourier series, we will continue with studying estimates for their 1-norms, where new phenomena occur. Among the results that will be discussed are Littlewood’s conjecture for harmonic Fourier series and Nazarov’s inequality for nonharmonic Fourier series.  

The format of the seminar is that of a reading seminar based on the paper “From Ingham to Nazarov’s inequality: a survey on some trigonometric inequalities” by P. Jaming and C. Saba.  

Reading list:
P. Jaming, C. Saba. From Ingham to Nazarov’s inequality: a survey on some trigonometric inequalities. Adv. Pure Appl. Math. 15, No. 3, 12-76 (2024). Link.