The book 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 book aims at clarifying 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.

The book is based on six lectures given by the author at the University of Michigan. This is reflected in the exposition, which is a blend of informal explanations with technical details.

The book is essentially self-contained. There is an underlying assumption that the reader has a basic knowledge of complex and functional analysis. Beyond that, the reader should have some familiarity with the basics of $H^p$ theory and BMO.

Readership

Graduate students and research mathematicians interested in analysis.