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The classical sampling problem is to reconstruct entire functions with given spectrum SS from their values on a discrete set LL. From the geometric point of view, the possibility of such reconstruction is equivalent to determining for which sets LL the exponential system with frequencies in LL forms a frame in the space L2(S)L2(S). The book also treats the problem of interpolation of discrete functions by analytic ones with spectrum in SS and the problem of completeness of discrete translates. The size and arithmetic structure of both the spectrum SS and the discrete set LL play a crucial role in these problems.

After an elementary introduction, the authors give a new presentation of classical results due to Beurling, Kahane, and Landau. The main part of the book focuses on recent progress in the area, such as construction of universal sampling sets, high-dimensional and non-analytic phenomena.

The reader will see how methods of harmonic and complex analysis interplay with various important concepts in different areas, such as Minkowski´s lattice, Kolmogorov´s width, and Meyer´s quasicrystals.

The book is addressed to graduate students and researchers interested in analysis and its applications. Due to its many exercises, mostly given with hints, the book could be useful for undergraduates.