One-sided Problems in Fourier Analysis

In the context of Fourier analysis on the real line, a \textit{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 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 his talk, Bartosz Malman will describe this mass condensation phenomenon and its applications to operator-theoretic problems.