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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.Read more
- The only book to cover signal processing on the unit sphere for applications in engineering and applied sciences such as geodesy and astronomy
- Includes historical anecdotes on important and influential people who contributed to the development of the theory
- Features over 100 worked problems, carefully chosen to make additional points, to motivate the reader and to deepen and broaden their understanding
- Explains the theory of reproducing kernel Hilbert spaces from a novel angle
Reviews & endorsements
"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, Department of Geosciences and Program in Applied and Computational Mathematics, Princeton UniversitySee more reviews
"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
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- Date Published: April 2013
- format: Hardback
- isbn: 9781107010031
- length: 440 pages
- dimensions: 244 x 170 x 25 mm
- weight: 0.9kg
- contains: 66 b/w illus. 15 tables 100 exercises
- availability: Available
Table of Contents
Part I. Hilbert spaces:
Part II. Operators:
3. Introduction to operators
4. Bounded operators
5. Compact operators
6. Integral operators
Part III. Applications:
7. Signals and systems analysis on the 2-sphere
8. Signal concentration and joint spatio-spectral analysis
9. Convolution on 2-sphere
10. Reproducing kernel Hilbert spaces.
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