Student Analysis Seminar

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On the pointwise convergence of Fourier series

Speaker: Robert Trosten, NYU Courant

Location: Warren Weaver Hall 517

Date: Tuesday, November 11, 2025, 11 a.m.

Synopsis:

When studying partial differential equations with periodic boundary conditions, Fourier series present a natural tool: you get an equation for the evolution of the Fourier coefficients, and sum back up to develop some representation of your function. How accurate is this representation? It turns out that this question is rather delicate – for example, if your function is merely integrable, this can diverge everywhere, while only a tiny bit more regularity ensures almost-everywhere convergence. This survey talk will be a broad yet deep dive into the pointwise convergence of Fourier series. We begin with a trip to the dark side, investigating when everywhere convergence fails – highlights include Baire category methods yielding failure for "most" continuous maps, and Kolmogorov's first papers yielding catastrophic failure for integrable functions. We will then take a turn to the light side of Fourier series convergence, building an understanding of the Wiener algebra of absolutely convergent Fourier series using both analytic and more algebraic methods. Finally, we will discuss the a.e. convergence of Fourier series: how much integrability do you need to guarantee a.e. convergence?