Bartosz Malman: One-sided problems in Fourier analysis

Abstract: In the context of Fourier analysis on the real line, a one-sided problem involves deducing properties of a function \(f\) from some information about the restriction of its Fourier transform \(\widehat{f}\) to a half-line, for instance to \(\mathbb{R}_- := (-\infty, 0)\). A prototypical result, which is foundational to the theory of Hardy spaces on \(\mathbb{R}\), asserts that if \(f \in L^2(\mathbb{R})\) is non-zero and \(\widehat{f}\) vanishes on a half-line, then \(f\) satisfies the \(\textit{Szeg\H{o} condition}\) \(\int_{-\infty}^\infty \frac{\log |f(x)|}{1+x^2} \, dx > -\infty\). Various problems in operator theory involve the study of functions \(f\) satisfying a weaker condition of decay of \(\widehat{f}\) on a half-line. In this setting, simple examples show that the Szeg\H{o} condition need not be satisfied. However, the following local \(\emph{Szeg\H{o}-type conditions}\) hold: if the decay of
\(\widehat{f}\) is strong enough on a half-line, then the mass of the function \(f \in L^2(\mathbb{R})\) must concentrate enough for the integral \(\int_E \log |f(x)| dx\) to converge on a "massive" set \(E\). In my talk, I will describe this mass condensation phenomenon and its applications to operator-theoretic problems.