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-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 $H^p$ theory and BMO.
General
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