Hilbert Space Methods In Signal Processing

Manufacturer

Cambridge University Press

This book is about the theory and applications of Hilbert spaces and dense sets. It starts by establishing the concept of countably infinite, which is central to the proper understanding of separable Hilbert spaces. Fundamental ideas such as convergence, completeness and dense sets are first demonstrated through simple familiar examples and then formalised. The book then goes on to cover the theory of bounded, compact and integral operators at an advanced but accessible level. Finally, the theory is put into action, considering signal processing on the unit sphere. The book is interspersed with historical comments about central figures in the development of the theory, which helps bring the subject to life.

This lively and accessible book describes the theory and applications of Hilbert spaces and also presents the history of the subject to reveal the ideas behind theorems and the human struggle that led to them. The authors begin by establishing the concept of 'countably infinite', which is central to the proper understanding of separable Hilbert spaces. Fundamental ideas such as convergence, completeness and dense sets are first demonstrated through simple familiar examples and then formalised. Having addressed fundamental topics in Hilbert spaces, the authors then go on to cover the theory of bounded, compact and integral operators at an advanced but accessible level. Finally, the theory is put into action, considering signal processing on the unit sphere, as well as reproducing kernel Hilbert spaces. The text is interspersed with historical comments about central figures in the development of the theory, which helps bring the subject to life. Review: 'A book of this mathematical sophistication shouldn't be this fun to read - or teach from! Guilty pleasure aside, the treatment of Hilbert spaces and operator theory is remarkable in its lucidity and completeness - several other textbooks' worth of material. More than half of the book consists of new insights into spherical data analysis cast in a general framework that will make any of us working in this and adjacent research areas reach for this book to properly understand what it is that we have done.' Frederik J. Simons, Princeton University 'The style is lively, and the mathematics is interspersed with historical remarks and anecdotes about the main mathematicians who developed the theory ... some insights are given that can [be] enlightening for professionals as well.' A. Blutheel, Mathematical Reviews