This volume is devoted to some topical problems and various applications of operator theory and its interplay with modern complex analysis. It consists of 30 carefully selected surveys and research papers.
The main subjects of the volume include:
· free interpolation by analytic functions in its development from the pathbreaking works by L. Carleson up to the most recent achievements and in its connections with the theory of singular integral operators and Carleson-type embedding theorems, moment problems etc.
· Szökefalvi-Nagy-Foias model spaces studied from the point of view of holomorphic spaces
· holomorphic spaces (Hardy, Bergman, Hölder, and Sobolev spaces)
· analytic functions smooth up to the boundary with their subtle properties related to the Nevanlinna-Smirnov factorization, division and multiplication, and zero sets
· a new approach to weighted inequalities for singular integrals based on the Bellman function in optimization theory;
· the uncertainty principle in harmonic analysis and, in particular, a complete version of Turan‘s lemma on trigonometric sums
· Hankel operators and stationary Gaussian processes
· Fourier multipliers, and spectral analysis of some differential operators.
These themes are united by the "operator theoretic ideology" and systematic use of modern function theoretical techniques.
The book is dedicated to the memory of S. A. Vinogradov. It contains a bibliographical note (with a lively portrait) of S. A. Vinogradov, a detailed survey of his mathematical achievements, and a complete list of publications, as well as the translations of two of Vinogradov‘s surveys whose Russian originals are now hardly accessible.
Stanislav Aleksandrovich Vinogradov, his life and mathematics.- List of publications of S.A.Vinogradov.- Interpolation problems for analytic functions continuous in the closed disk and for functions whose sequence of coefficients is inlp.- Free interpolation in spaces of analytic functions.- Contributed Papers.- On embedding theorems for coinvariant subspaces of the shift operator, I.- Continuous and compact embeddings between star-invariant subspaces.- Rational functions in Bergman spaces.- S. A. Vinogradov, as I remember him.- The Bellman functions and sharp weighted inequalities for square functions.- Multiplicative chaos and multimeasures.- Some remarks to problems of approximation with prescribed rate.- Interpolation involving bounded bianalytic functions.- Carleson’s interpolation theorem deduced from a result of Pick.- A–?zero sets: new methods and techniques.- Interpolation sets for the Hölder spaces of functions analytic in a strip.- Scattering problem with physical behavior of scattering matrix and operator relations.- Spectra of inner functions andlp-multipliers.- The theorem on three spheres for harmonic differential forms.- Traces and extensions of multipliers in pairs of Sobolev spaces.- Complete version of Turan’s lemma for trigonometric polynomials on the unit circumference.- Ball, Haagerup, and distribution functions.- Carleson measures of Bergman spaces in domains with nonsmooth boundary.- On the zeros of tails of power series.- Regularity conditions for vectorial stationary processes.- The Friedrichs operator of a planar domain.- Parametrical representations of some classes of holomorphic functions in the disk.- Double power series and reproducing kernels.- Outer functions in yet another class of analytic functions.- Estimates for the approximation numbers of the weighted Riemann-Liouville operator in the spacesLp.- Special transformations of Cauchy type integral spaces.- A dimension-free Carleson measure inequality.- Carleman formula for some spaces of functions analytic in the disk and smooth in its closure.
