Foundations Of Time Frequency Analysis

In this chapter we discuss the argument principle and develop several of its consequences. In Section 1 we derive the argument principle from the residue theorem, and we use the argument principle to locate the zeros of analytic functions. Sections 2 through 5 can be viewed as a study of how the zeros of an analytic function depend on various types of parameters. Sections 6 and 7 are devoted to winding numbers of closed paths and the jump theorem for the Cauchy integral. The jump theorem yields an easy proof of the Jordan curve theorem in the smooth case, and a proof of the full Jordan curve theorem is laid out in the exercises. In Section 8 we introduce simply connected domains and we characterize these in several ways. While the material in this chapter is of fundamental importance for the Riemann mapping theorem in Chapter XI and for various further developments, the student can skip to Chapter IX immediately after Sections 1 and 2.

The Logarithmic Integral

An H(b) space is defined as a collection of analytic functions which are in the image of an operator. The theory of H(b) spaces bridges two classical subjects: complex analysis and operator theory, which makes it both appealing and demanding. The first volume of this comprehensive treatment is devoted to the preliminary subjects required to understand the foundation of H(b) spaces, such as Hardy spaces, Fourier analysis, integral representation theorems, Carleson measures, Toeplitz and Hankel operators, various types of shift operators, and Clark measures. The second volume focuses on the central theory. Both books are accessible to graduate students as well as researchers: each volume contains numerous exercises and hints, and figures are included throughout to illustrate the theory. Together, these two volumes provide everything the reader needs to understand and appreciate this beautiful branch of mathematics.

The Theory of H(b) Spaces

Harmonic Measure: Jordan Domains

Weighted discrete Hilbert transforms (an)n 㜡 P n anvn=(z − n) from ` 2 to a weighted L 2 space are studied, with = ( n) a sequence of distinct points in the complex plane and v = (vn) a corresponding sequence of positive numbers. In the special case when j nj grows at least exponentially, bounded transforms of this kind are described in terms of a simple relative to the Muckenhoupt (A2) condition. The special case when z is restricted to another sequenceis studied in detail; it is shown that a bounded trans- form satisfying a certain admissibility condition can be split into finitely many surjective transforms, and precise geometric conditions are found for invertibility of such two weight transforms. These results can be interpreted as statements about systems of reproducing kernels in certain Hilbert spaces of which de Branges spaces and model subspaces of H 2 are prime examples. In particular, a connection to the Feichtinger conjecture is pointed out. Descriptions of Carleson measures and Riesz bases of normalized reproducing kernels for certain "small" de Branges spaces follow from the results of this paper. 1. Introduction and main results This paper is concerned with the mapping properties of what we will call discrete Hilbert transforms in the complex plane. One aspect of this topic was treated in (7), where all unitary discrete Hilbert transforms were described. When we now turn to questions about boundedness, surjectivity, and invertibility, results of the same generality seem at present out of reach. The results to be presented below are complete only when the discrete Hilbert transforms are defined on particularly sparse sequences. We will nevertheless present the problems in the most general setting, as we believe they merit further investigations. We will emphasize the connection with topics such as Carleson measures and Riesz bases of normalized reproducing kernels in Hilbert spaces of analytic functions; this will lead us to the most intriguing general question, namely whether or not the Feichtinger conjecture holds true for systems of reproducing kernels in such spaces. We begin by assuming that we are given an infinite sequence of distinct points = ( n) in C and a corresponding sequence of positive numbers v = (vn), both indexed by the

Discrete Hilbert transforms on sparse sequences

Weighted discrete Hilbert transforms $(a_n)_n \mapsto \sum_n a_n v_n/(z-\gamma_n)$ from $\ell^2_v$ to a weighted $L^2$ space are studied, with $\Gamma=(\gamma_n)$ a sequence of distinct points in the complex plane and $v=(v_n)$ a corresponding sequence of positive numbers. In the special case when $|\gamma_n|$ grows at least exponentially, bounded transforms of this kind are described in terms of a simple relative to the Muckenhoupt $(A_2)$ condition. The special case when $z$ is restricted to another sequence $\Lambda$ is studied in detail; it is shown that a bounded transform satisfying a certain admissibility condition can be split into finitely many surjective transforms, and precise geometric conditions are found for invertibility of such two weight transforms. These results can be interpreted as statements about systems of reproducing kernels in certain Hilbert spaces of which de Branges spaces and model subspaces of $H^2$ are prime examples. In particular, a connection to the Feichtinger conjecture is pointed out. Descriptions of Carleson measures and Riesz bases of normalized reproducing kernels for certain "small" de Branges spaces follow from the results of this paper.

Hereditary completeness for systems of exponentials and reproducing kernels

We solve completely the spectral synthesis problem for reproducing kernels in the de Branges spaces $${\mathcal{H}(E)}$$
 . Namely, we describe the de Branges spaces $${\mathcal{H}(E)}$$
 such that every complete and minimal system of reproducing kernels $${\{k_\lambda\}_{\lambda \in \Lambda}}$$
 with complete biorthogonal $${\{g_\lambda\}_{\lambda \in \Lambda}}$$
 admits the spectral synthesis, i.e., $${f \in \overline{\rm Span} \{(f, g_\lambda) k_\lambda : \lambda \in \Lambda\}}$$
 for any f in $${\mathcal{H}(E)}$$
 . Surprisingly, this property takes place only for two essentially different classes of de Branges spaces: spaces with finite spectral measure and spaces which are isomorphic to Fock-type spaces of entire functions. The first class goes back to de Branges himself, while special cases of de Branges spaces of the second class appeared in the literature only recently; we give a complete characterisation of this second class in terms of the spectral data for $${\mathcal{H}(E)}$$
 .

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Spectral synthesis in de Branges spaces

We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials $\{e^{i\lambda_n t}\}$ in $L^2(-a,a)$ is hereditarily complete up to a one-dimensional defect. This means that there is at most one (up to a constant factor) function $f$ which is orthogonal to all the summands in its formal Fourier series $\sum_n (f,\tilde e_n) e^{i\lambda_n t}$, where $\{\tilde e_n\}$ is the system biorthogonal to $\{e^{i\lambda_n t}\}$. However, this one-dimensional defect is possible and, thus, there exist nonhereditarily complete exponential systems. Analogous results are obtained for systems of reproducing kernels in de Branges spaces. For a wide class of de Branges spaces we construct nonhereditarily complete systems of reproducing kernels, thus answering a question posed by N. Nikolski.

Yurii Belov

Papers

Discrete Hilbert transforms on sparse sequences

Discrete Hilbert transforms on sparse sequences

Hereditary completeness for systems of exponentials and reproducing kernels

Spectral synthesis in de Branges spaces

Hereditary completeness for systems of exponentials and reproducing kernels