We introduce and study a family of spaces of entire functions in one variable that generalise the classical Paley-Wiener and Bernstein spaces. Namely, we consider entire functions of exponential type $a$ whose restriction to the real line belongs to the homogeneous Sobolev space $\dot{W}^{s,p}$ and we call these spaces fractional Paley-Wiener if $p=2$ and fractional Bernstein spaces if $p\in(1,\infty)$, that we denote by $PW^s_a$ and $\mathcal B^{s,p}_a$, respectively. For these spaces we provide a Paley-Wiener type characterization, we remark some facts about the sampling problem in the Hilbert setting and prove generalizations of the classical Bernstein and Plancherel-Pólya inequalities. We conclude by discussing a number of open questions.
Let $S_1$ and $S_2$ be disjoint finite unions of parallelepipeds. We describe necessary and sufficient conditions on the sets $S_1,S_2$ and exponents $p$ such that the canonical projection $P$ from $PW_{S_1\cup S_2}^p$ to $PW_{S_1}^p$ is a contraction.
For a sequence of complex numbers $Λ$ we consider the restriction operator $R_Λ$ defined on Paley-Wiener spaces $PW_τ^{p}$ ($1<p<\infty$). Lyubarskii and Seip gave necessary and sufficient conditions on $Λ$ for $R_Λ$ to be an isomorphism between $PW_τ^{p}$ and a certain weighted $l^{p}$ space. The Carleson condition appears to be necessary. We extend their result to $N-$Carleson sequences (finite unions of $N$ disjoint Carleson sequences). More precisely, we give necessary and sufficient conditions for $R_Λ$ to be an isomorphism between $PW_τ^{p}$ and an appropriate sequence space involving di…
In the Clifford algebra setting the present study develops three reproducing kernel Hilbert spaces of the Paley-Wiener type, namely the Paley-Wiener spaces, the Hardy spaces on strips, and the Bergman spaces on strips. In particular, we give spectrum characterizations and representation formulas of the functions in those spaces and estimation of their respective reproducing kernels.
The present work reviews the second half of Lyubarskii and Seip's paper, Weighted Paley--Wiener Spaces. Axioms defining a larger class of de Branges spaces are abstracted, allowing us to state and prove their results at a higher level of generality.
For a fixed singular Borel probability measure $μ$ on $\mathbb{T}$, we give several characterizations of when an entire function is the Fourier transform of some $f \in L^2(μ)$. The first characterization is given in terms of criteria for sampling functions of the form $\hat{f}$ when $f \in L^2(μ)$. The second characterization is given in terms of criteria for interpolation of bounded sequences on $\mathbb{N}_{0}$ by $\hat{f}$. Both characterizations use the construction of Fourier series for $f \in L^2(μ)$ demonstrated in Herr and Weber via the Kaczmarz algorithm and classical results concern…
Starting from a remark about the computation of Kashiwara-Schapira's enhanced Laplace transform by using the Dolbeault complex of enhanced distributions, we explain how to obtain explicit holomorphic Paley-Wiener-type theorems. As an example, we get back some classical theorems due to Polya and Méril as limits of tempered Laplace-isomorphisms. In particular, we show how contour integrations naturally appear in this framework.
In this paper we study spaces of holomorphic functions on the Siegel upper half-space $\mathcal U$ and prove Paley-Wiener type theorems for such spaces. The boundary of $\mathcal U$ can be identified with the Heisenberg group $\mathbb H_n$. Using the group Fourier transform on $\mathbb H_n$, Ogden-Vagi proved a Paley-Wiener theorem for the Hardy space $H^2(\mathcal U)$. We consider a scale of Hilbert spaces on $\mathcal U$ that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury-Arveson space, and the Dirichlet space $\mathcal D$. F…
We give a general inequality for Bergman kernels of Bergman spaces defined by convex weights in $\C^n$. We also discuss how this can be used in Nazarov's proof of the Bourgain-Milman theorem, as a substitute for Hörmander's estimates for the $\dbar$-equation.
It is well known from a result by Shapiro-Shields that in the Hardy spaces, a sequence of reproducing kernels is uniformly minimal if and only if it is an unconditional basis in its span. This property which can be reformulated in terms of interpolation and so-called weak interpolation is not true in Paley-Wiener spaces in general. Here we show that the Carleson condition on a sequence $Λ$ together with minimality in Paley-Wiener spaces $PW_τ^{p}$ of the associated sequence of reproducing kernels implies the interpolation property of $Λ$ in $PW_{τ+ε}^{p}$, for every $ε>0$. With the same techni…
In the classical phase retrieval problem in the Paley-Wiener class $PW_L$ for $L>0$, i.e. to recover $f\in PW_L$ from $|f|$, Akutowicz, Walther, and Hofstetter independently showed that all such solutions can be obtained by flipping an arbitrary set of complex zeros across the real line. This operation is called zero-flipping and we denote by $\mathfrak{F}_a f$ the resulting function. The operator $\mathfrak{F}_a$ is defined even if $a$ is not a genuine zero of $f$, that is if we make an error on the location of the zero. Our main goal is to investigate the effect of $\mathfrak{F}_a$. We show …
We consider the problem of conjugate phase retrieval in Paley-Wiener space $PW_π$. The goal of conjugate phase retrieval is to recover a signal $f$ from the magnitudes of linear measurements up to unknown phase factor and unknown conjugate, meaning $f(t)$ and $\overline{f(t)}$ are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in $PW_π$ by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in $PW_π$ by sampling both $f$ and $f^{\prime}$ only on the …
In this paper, we show that the expansions of functions from $L^p$-Paley-Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1<p<\infty$, even in the cases when they might not converge in $L^p$-norm. We thereby consider the classical Paley-Wiener spaces $PW_c^p\subset L^p(\mathcal{R})$ of functions whose Fourier transform is supported in $[-c,c]$ and Paley-Wiener like spaces $B_{α,c}^p\subset L^p(0,\infty)$ of functions whose Hankel transform $\mathcal{H}^α$ is supported in $[0,c]$.As a side product, we show the continuity of the projection oper…
The celebrated Paley-Wiener theorem naturally identifies the spaces of bandlimited functions with subspaces of entire functions of exponential type. Recently, it has been shown that these spaces remain invariant only under composition with affine maps. After some motivation demonstrating the importance of characterization of range spaces of bandlimited functions, in this paper we identify the subspaces of $ L^2 (\mathbb{R}) $ generated by these action. Extension of these theorems where Paley-Wiener spaces are replaced by the deBranges-Rovnyak spaces are given.
We discuss relations between uniform minimality, unconditionality and interpolation for families of reproducing kernels in backward shift invariant subspaces. This class of spaces contains as prominent examples the Paley-Wiener spaces for which it is known that uniform minimality does in general neither imply interpolation nor unconditionality. Hence, contrarily to the situation of standard Hardy spaces (and other scales of spaces), changing the size of the space seems %in this context necessary to deduce unconditionality or interpolation from uniform minimality. Such a change can take two dir…
We prove that for every radial weighted Fock space, the system biorthogonal to a complete and minimal system of reproducing kernels is also complete under very mild regularity assumptions on the weight. This result generalizes a theorem by Young on reproducing kernels in the Paley--Wiener space and a recent result of Belov for the classical Bargmann--Segal--Fock space.
Extending earlier work by Rogers, Wiener-Hopf factorisation is studied for a class of functions closely related to Nevanlinna-Pick functions and complete Bernstein functions. The name 'Rogers functions' is proposed for this class. Under mild additional condition, for a Rogers function f, the Wiener--Hopf factors of f(z)+q, as well as their ratios, are proved to be complete Bernstein functions in both z and q. This result has a natural interpretation in fluctuation theory of Lévy processes: for a Lévy process X_t with completely monotone jumps, under mild additional condition, the Laplace expon…
The Hörmander--Bernhardsson function in higher dimensions
IMPA - Instituto de Matemática Pura e Aplicada（巴西纯粹与应用数学研究所）；Institut des Hautes Études Scientifiques, CNRS, Laboratoire Alexander Grothendieck（高等科学研究所）
The de Branges spaces of entire functions generalise the classical Paley-Wiener space of square summable bandlimited functions. Specifically, the square norm is computed on the real line with respect to weights given by the values of certain entire functions. For the Paley-Wiener space, this can be chosen to be an exponential function where the phase increases linearly. As our main result, we establish a natural geometric characterisation, in terms of densities, for real sampling and interpolating sequences in the case when the derivative of the phase function merely gives a doubling measure o…
