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Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering

Ole Christensen
Publication Date: 
Number of Pages: 
Applied and Numerical Harmonic Analysis
[Reviewed by
Allen Stenger
, on

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 Lp 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 Lp 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, a math help site that fosters inquiry learning.

  • ANHA Series Preface
  • Preface
  • Prologue
  1. Mathematical Background
    1. Rn and Cn
    2. Abstract vector spaces
    3. Finite-dimensional vector spaces
    4. Topology in Rn
    5. Supremum and infimum
    6. Continuity of functions on R
    7. Integration and summation
    8. Some special functions
    9. A useful technique: proof by induction
    10. Exercises
  2. Normed Vector Spaces
    1. Normed vector spaces
    2. Topology in normed vector spaces
    3. Approximation in normed vector spaces
    4. Linear operators on normed spaces
    5. Series in normed vector spaces
    6. Exercises
  3. Banach Spaces
    1. Banach spaces
    2. The Banach spaces l1(N) and l p(N)
    3. Linear operators on Banach spaces
    4. Exercises
  4. Hilbert Spaces
    1. Inner product spaces
    2. The Hilbert space l2(N)
    3. Orthogonality and direct sum decomposition
    4. Functionals on Hilbert spaces
    5. Linear operators on Hilbert spaces.
    6. Bessel sequences in Hilbert spaces
    7. Orthonormal bases
    8. Frames in Hilbert spaces
    9. Exercises
  5. The Lp-spaces
    1. Vector spaces consisting of continuous functions
    2. The vector space L1(R)
    3. Integration in L1(R)
    4. The spaces Lp(R)
    5. The spaces Lp(a,b)
    6. Exercises
  6. The Hilbert Space L2
    1. The Hilbert space L2(R)
    2. Linear operators on L2(R)
    3. The space L2(a,b)
    4. Fourier series revisited
    5. Exercises
  7. The Fourier Transform
    1. The Fourier transform on L1(R)
    2. The Fourier transform on L2(R)
    3. Convolution
    4. The sampling theorem
    5. The discrete Fourier transform
    6. Exercises
  8. An Introduction to Wavelet Analysis
    1. Wavelets
    2. Multiresolution analysis
    3. Vanishing moments and the Daubechies’ wavelets
    4. Wavelets and signal processing
    5. Exercises
  9. A Closer Look at Multiresolution Analysis
    1. Basic properties of multiresolution analysis
    2. The spaces Vj and Wj
    3. Proof of Theorem 8.2.7
    4. Proof of Theorem 8.2.11
    5. Exercises
  10. B-splines
    1. The B-splines Nm
    2. The centered B-splines Bm
    3. B-splines and wavelet expansions
    4. Frames generated by B-splines
    5. Exercises
  11. Special Functions
    1. Regular Sturm–Liouville problems
    2. Legendre polynomials
    3. Laguerre polynomials
    4. Hermite polynomials
    5. Exercises
  • Appendix A
    1. A.1 Proof of Weierstrass’ theorem, Theorem 2.3.4
    2. A.2 Proof of Theorem 7.1.7
    3. A.3 Proof of Theorem 10.1.5
    4. A.4 Proof of Theorem 11.2.2
  • Appendix B
    1. B.1 List of vector spaces
    2. B.2 List of special polynomials
  • List of Symbols
  • References
  • Index