Interpolation and Sampling in Spaces of Analytic FunctionsAmerican Mathematical Soc., 2004 - 139 páginas This book contains the latest developments in a central theme of research on analysis of one complex variable. The material is based on lectures at the University of Michigan. The exposition is about understanding the geometry of interpolating and sampling sequences in classical spaces of analytic functions. The subject can be viewed as arising from three classical topics: Nevanlinna-Pick interpolation, Carleson's interpolation theorem for $Hinfty$, and the sampling theorem, also known as the Whittaker-Kotelnikov-Shannon theorem. The author clarifies how certain basic properties of the space at hand are reflected in the geometry of interpolating and sampling sequences. Key words for the geometric descriptions are Carleson measures, Beurling densities, the Nyquist rate, and the Helson-Szego condition. Seip writes in a relaxed and fairly informal style, successfully blending informal explanations with technical details. The result is a very readable account of this complex topic. Prerequisites are a basic knowledge of complex and functional analysis. Beyond that, readers should have some familiarity with the basics of $Hp$ theory and BMO. |
Contenido
Chapter 1 Carlesons interpolation theorem | 1 |
Chapter 2 Interpolating sequences and the Pick property | 15 |
Chapter 3 Interpolation and sampling in Bergman spaces | 41 |
Chapter 4 Interpolation in the Bloch space | 63 |
Chapter 5 Interpolation sampling and Toeplitz operators | 79 |
Chapter 6 Interpolation and sampling in PaleyWiener spaces | 95 |
| 125 | |
| 135 | |
Back Cover | 140 |
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Términos y frases comunes
analytic functions arbitrary argument assume Bergman spaces Beurling Blaschke product Bloch space Bøe Carleson condition Carleson measure Carleson measure condition Carleson's theorem Cauchy–Schwarz inequality Chapter complete interpolating sequence conformal invariance consider constant Corollary corona theorem defined denote Dirichlet space distinct points dyadic eacist entire functions equivalent estimate exists fe H follows Fourier function f Helson–Szegó condition Hilbert space implies inner function interpolating Blaschke product interpolation and sampling interpolation problem f(z Landau's Lemma multiplier algebra Nevanlinna–Pick norm obtain orthogonal Paley–Wiener space Paley–Wiener theorem Paragraph Pick property Pick's theorem proof Proposition prove real line reproducing kernels result Riesz sampling sequence sampling theorem satisfies the Carleson separated sequence sequence for PW Shapiro and Shields shows ſº solution space of analytic subharmonic function subspace Toeplitz operator uniformly bounded universal interpolating sequences upper half-plane zero sequence