Blaschke Products and Their Applications

This is a selection of papers presented at the the conference Blaschke products and their applications held at the Fields Institute in Toronto July 25-29, 2011. It is volume 65 of the Series Fields Institute Communications.

Blaschke products were introduced by Blaschke in 1915 to study analytic functions in the complex unit disk. These (infinite) products are examples of inner functions, i.e., functions analytic in the unit disk with boundary value 1 a.e. A similar definition exists for functions analytic in the upper half plane. Blaschke products may collect the zeros of of functions in Hardy spaces and are therefore an essential element of inner-outer factorization of these functions. Blaschke functions and their derivatives appeared in several applications like approximation theory, solution of extremal problems, operator theory, differential equations, geometry, and even computability theory. Some of these applications are treated in the contributions of these proceedings. Among the fifteen papers are a few survey papers which are usually longer, but most of them are shorter research-type papers. The titles with a one-sentence description are as follows. Even if the title does not mention it explicitly, all these papers involve Blaschke products one way or another.

  1. Applications of Blaschke Products to the Spectral Theory of Toeplitz Operators (30 pp.) (Sergei Grudsky and Eugene Shargorodsky) A survey-type paper with some open problems.
  2. Approximating the Riemann Zeta-Function by Strongly Recurrent Functions (12 pp.) (P.M. Gauthier) Research paper. The approximant shares properties of the zeta-function.
  3. A Survey on Blaschke-Oscillatory Differential Equations, with Updates (56 pp.) (Janne Heittokangas) A survey of the study of the differential equation f"+A(z)f = 0, with A analytic and its generalizations.
  4. Bi-orthogonal Expansions in the Space L²(0,∞) (14 pp.) (Andre Boivin and Changzhong Zhu) Research paper about expanding a function in an exponential basis and its biorthogonal counterpart.
  5. Blaschke Products as Solutions of a Functional Equation (6 pp.) (Javad Mashreghi) Study of the solution of ψ(φ(z))s = ψ(z) with φ given, but ψ unknown. The problem is not completely solved.
  6. Cauchy Transforms and Univalent Functions (13 pp.) (Joseph A. Cima and John A. Pfaltzgraff) Study of Cauchy transforms of a measure on the unit circle which give univalent functions.
  7. Critical Points, the Gauss Curvature Equation and Blaschke Products (25 pp.) (Daniela Kraus and Oliver Roth) A survey of results that are known about points where the derivative of a Blaschke product vanishes and the functions having these as zeros.
  8. Growth, Zero Distribution and Factorization of Analytic Functions of Moderate Growth in the Unit Disc (15 pp.) (Igor Chyzhykov and Severyn Skaskiv) This is a short survey about the topic described by the title.
  9. Hardy Means of a Finite Blaschke Product and Its Derivative (12 pp.) (Alan Gluchoff and Frederick Hartmann) A research paper with a conjecture.
  10. Hyperbolic Derivatives Determine a Function Uniquely (6 pp.) (Line Baribeau) It proves that the sequence of hyperbolic derivatives defines a function mapping the unit disc to itself.
  11. Hyperbolic Wavelets and Multiresolution in the Hardy Space of the Upper Half Plane (16 pp.) (Hans G. Feichtinger and Margit Pap) A rational orthogonal wavelet is constructed generating a multiresolution for H² of the upper half plane.
  12. Norms of Composition Operators Induced by Finite Blaschke Products on Möbius Invariant Spaces (14 pp.) (María J. Martín and Dragan Vukokotić) The norms of these operators on Bloch and Dirichlet spaces are computed.
  13. On the Computable Theory of Bounded Analytic Functions (26 pp.) (Timothy H. McNicholl) The theory of bounded analytic functions is reexamined from the viewpoint of computability theory.
  14. Polynomials Versus Finite Blaschke Products (26 pp.) (Tuen Wai Ng and Chiu Yin Tsang) This research paper shows analogy of properties between (Chebyshev) polynomials and (Chebyshev) finite Blaschke products.
  15. Recent Progress on Truncated Toeplitz Operators (45 pp.) (Stephan Ramon Garcia and William T. Ross) Another survey paper giving the advances in the theory of Toeplitz operators since 2007.

The book is of interest both to students and researchers who need the theory of complex functions and especially those who are working in Hardy and related spaces.

A. Bultheel
KU Leuven
Book details

This contains 15 (survey and research) papers presented at the conference Blaschke products and their applications held at the Fields Institute in Toronto July 25-29, 2011.

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