Publisher:

Birkhäuser

Number of Pages:

263

Price:

49.95

ISBN:

9780817649791

This book is a precariously-positioned and choppy introduction to function spaces and orthogonal expansions, aimed at students of applied mathematics and engineering. Its position is precarious because it describes a large number of tools without showing any of them in action. Because of this omission, it reads like the first book (“Theory”) of a two-volume set, for which we don’t have the “Applications” volume. It is choppy because each (short) chapter is largely independent of the others; there’s no real thread connecting them.

In the book’s favor, it is clearly written and it does provide a useful summary of the basic properties of the tools it covers. It does a good job of explaining the difference in the various function spaces. It has detailed coverage of wavelets and the related subjects of B-splines and multiresolution analysis, although still without applications.

Although it’s not a cookbook, it doesn’t give a complete set of proofs either. It tends to prove only the easier theorems, stating the more difficult ones without proof. The book has about 150 exercises, and most of these do not test the student‘s mastery or advance the narrative but merely complete proofs that were sketched in the body. The book undertakes the unenviable task of explaining the L^{p} spaces without explicitly using the Lebesgue integral, even though it does quote most of the key theorems of Lebesgue theory, including the bounded and dominated convergence theorems and the completeness of the L^{p} spaces.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

Date Received:

Monday, June 28, 2010

Reviewable:

Yes

Series:

Applied and Numerical Harmonic Analysis

Publication Date:

2010

Format:

Hardcover

Audience:

Category:

Textbook

Allen Stenger

12/6/2010

- ANHA Series Preface
- Preface
- Prologue

- Mathematical Background
- R
^{n}and C^{n} - Abstract vector spaces
- Finite-dimensional vector spaces
- Topology in R
^{n} - Supremum and infimum
- Continuity of functions on R
- Integration and summation
- Some special functions
- A useful technique: proof by induction
- Exercises

- R
- Normed Vector Spaces
- Normed vector spaces
- Topology in normed vector spaces
- Approximation in normed vector spaces
- Linear operators on normed spaces
- Series in normed vector spaces
- Exercises

- Banach Spaces
- Banach spaces
- The Banach spaces
*l*^{1}(N) and*l*^{ p}(N) - Linear operators on Banach spaces
- Exercises

- Hilbert Spaces
- Inner product spaces
- The Hilbert space
*l*^{2}(N) - Orthogonality and direct sum decomposition
- Functionals on Hilbert spaces
- Linear operators on Hilbert spaces.
- Bessel sequences in Hilbert spaces
- Orthonormal bases
- Frames in Hilbert spaces
- Exercises

- The L
^{p}-spaces- Vector spaces consisting of continuous functions
- The vector space L
^{1}(R) - Integration in L
^{1}(R) - The spaces L
^{p}(R) - The spaces L
^{p}(a,b) - Exercises

- The Hilbert Space L
^{2}- The Hilbert space L
^{2}(R) - Linear operators on L
^{2}(R) - The space L
^{2}(a,b) - Fourier series revisited
- Exercises

- The Hilbert space L
- The Fourier Transform
- The Fourier transform on L
^{1}(R) - The Fourier transform on L
^{2}(R) - Convolution
- The sampling theorem
- The discrete Fourier transform
- Exercises

- The Fourier transform on L
- An Introduction to Wavelet Analysis
- Wavelets
- Multiresolution analysis
- Vanishing moments and the Daubechies’ wavelets
- Wavelets and signal processing
- Exercises

- A Closer Look at Multiresolution Analysis
- Basic properties of multiresolution analysis
- The spaces V
_{j}and W_{j} - Proof of Theorem 8.2.7
- Proof of Theorem 8.2.11
- Exercises

- B-splines
- The B-splines N
_{m} - The centered B-splines B
_{m} - B-splines and wavelet expansions
- Frames generated by B-splines
- Exercises

- The B-splines N
- Special Functions
- Regular Sturm–Liouville problems
- Legendre polynomials
- Laguerre polynomials
- Hermite polynomials
- Exercises

- Appendix A
- A.1 Proof of Weierstrass’ theorem, Theorem 2.3.4
- A.2 Proof of Theorem 7.1.7
- A.3 Proof of Theorem 10.1.5
- A.4 Proof of Theorem 11.2.2

- Appendix B
- B.1 List of vector spaces
- B.2 List of special polynomials

- List of Symbols
- References
- Index

Publish Book:

Modify Date:

Monday, December 6, 2010

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