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Register a new userExtreme points of the unit ball of Paley-Wiener space over two symmetric intervals
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Let $PW_S^1$ be the space of integrable functions on $mathbb{R}$ whose Fourier transform vanishes outside $S$, where $S = [-sigma,-rho]cup[rho,sigma]$, $0<rho<sigma$. In the case $rho>sigma/2$, we present a complete description of the extreme points of the unit ball of $PW_S^1$. This description is no longer true if $rho<sigma/2$. For $rho>sigma/2$ we also show that every $f in PW^1_S, , |f|_1 =1,$ can be represented as $f = (f_1 + f_2)/2$ where $f_1$ and $f_2$ are extreme.
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We develop real Paley-Wiener theorems for classes ${mathcal S}_omega$ of ultradifferentiable functions and related $L^{p}$-spaces in the spirit of Bang and Andersen for the Schwartz class. We introduce results of this type for the so-called Gabor transform and give a full characterization in terms of Fourier and Wigner transforms for several variables of a Paley-Wiener theorem in this general setting, which is new in the literature. We also analyze this type of results when the support of the function is not compact using polynomials. Some examples are given.
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We establish the necessary and sufficient conditions for those symbols $b$ on the Heisenberg group $mathbb H^{n}$ for which the commutator with the Riesz transform is of Schatten class. Our main result generalises classical results of Peller, Janson--Wolff and Rochberg--Semmes, which address the same question in the Euclidean setting. Moreover, the approach that we develop bypasses the use of Fourier analysis, and can be applied to characterise that the commutator is of the Schatten class in other settings beyond Euclidean.
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