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Interpolation and Sampling in Spaces of Analytic Functions
Kristian Seip, Norwegian University of Science and Technology, Trondheim, Norway
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University Lecture Series
2004; 139 pp; softcover
Volume: 33
ISBN-10: 0-8218-3554-8
ISBN-13: 978-0-8218-3554-8
List Price: US$32
Member Price: US$25.60
Order Code: ULECT/33
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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 \(H^\infty\), 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-Szegő 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 \(H^p\) theory and BMO.

Readership

Graduate students and research mathematicians interested in analysis.

Table of Contents

  • Carleson's interpolation theorem
  • Interpolating sequences and the Pick property
  • Interpolation and sampling in Bergman spaces
  • Interpolation in the Bloch space
  • Interpolation, sampling, and Toeplitz operators
  • Interpolation and sampling in Paley-Wiener spaces
  • Bibliography
  • Index
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