The Use of Frames in Sampling Theory

This book provides a comprehensive review of regular sampling based on frame theory in a separable Hilbert space. Thus, sampling theory has common features in almost all situations: classical theory, Kramer sampling theory, and finite sampling or sampling Hilbert–Schmidt operators. In addition, the transversality of sampling theory with other mathematical fields appears, in an easy way. The first three chapters of the book can be used as an introduction to sampling theory, while the rest of the chapters are addressed to introduce the interested reader in the research on the topic.

- What does sampling theory mean?.- Basic sampling theory.- Sampling in shift-invariant subspaces.- A review on Kramer sampling theorem.- A generalized sampling theory.- Finite frames related to sampling in finite-dimensional U-invariant subspaces.- Sampling in shift-invariant-like subspaces of Hilbert-Schmidt operators.

Antonio García García was born in Valladolid, Spain, in 1957. He received M.Sc. degree in mathematics in 1979 and a Ph.D. in mathematics in 1983, both from the University of Valladolid, Spain. After teaching mathematics in a high school, he joined, in 1991, the Department of Mathematics of the University Carlos III de Madrid, Spain, as an Associate Professor. Since 2009, he has held a full professor position. His research interests include all the mathematical topics related to sampling theory, where he has 6 Ph.D. students, and he has published 71 papers in periodic journals and 5 book chapters (4 in Springer books). H has also written 2 textbooks (in Spanish). He was the main researcher of 5 research projects on the sampling topic. More information can be found on his personal webpage.