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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 42: Fourier analysis

Introduction

Fourier analysis studies approximations and decompositions of functions using trigonometric polynomials. Of incalculable value in many applications of analysis, this field has grown to include many specific and powerful results, including convergence criteria, estimates and inequalities, and existence and uniqueness results. Extensions include the theory of singular integrals, Fourier transforms, and the study of the appropriate function spaces. This heading also includes approximations by other orthogonal families of functions, including orthogonal polynomials and wavelets.

History

Applications and related fields

Subfields

• 42A: Fourier analysis in one variable
• 42B: Fourier analysis in several variables, For automorphic theory, see mainly 11F30
• 42C: Nontrigonometric Fourier analysis

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

• Zygmund
• Strichartz, Robert S.: "A guide to distribution theory and Fourier transforms", Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1994. x+213 pp. ISBN 0-8493-8273-4 MR95f:42001
• Online introduction on Fourier analysis [Forrest Hoffman]
• Kaiser, Gerald: "A friendly guide to wavelets", Birkhäuser Boston, Inc., Boston, MA, 1994. 300 pp. ISBN 0-8176-3711-7 MR95i:94003
• Online tutorials on wavelets by IEEE and Robi Polikar
• There is a SIAM Activity Group on Orthogonal Polynomials and Special Functions. Their web site points to the OP-SF NET (electronic newsletter), preprint servers, and other professional concerns.
• The Wavelet Digest has a home page with back issues and related links.
• Duhamel, P.; Vetterli, M.: "Fast Fourier transforms: a tutorial review and a state of the art", Signal Process. 19 (1990), no. 4, 259--299. MR91a:94004

Software and tables

• Orthogonal polynomials software
• IBM Engineering and Scientific Subroutine Library
• Packages for Mathematica, versions 2.2 and 3.0.

Other web sites with this focus

Web Resources for Harmonic Analysis (Terry Tao).

• Several pages on wavelets:
  • Amara's Wavelets Page
  • Wavelets
  • Wavelets at SIAM.
• UTK archives page
• Here are the AMS and Goettingen resource pages for area 42.

Selected Topics at this site

• How are Fourier Transforms used?
• Statement of Fourier's theorem (representation by series)
• Determining Fourier transform with Hermite functions
• Suggested method to compensate for Gibbs' phenomenon
• Example of Chebyshev polynomial expansion (for ln(x+1) on [0,20])
• Products of Hermite polynomials as sums of Hermite polynomials
• Proving orthonormality of Legendre polynomials
• Orthogonal polynomials satisfy coupled 2-term recurrence relations
• Using Fourier analysis to characterize the sine function.
• Some suggestions for implementations of Fast Fourier Transforms.
• Variation among implementations of the Fast Fourier Transform.
• Code and references for Hankel transforms.
• How wavelets developed into tools for image compression (etc.): nice book review by Peter A. McCoy of Yves Meyer's book.
• Overview of operations with wavelets
• What's new in Singular Integrals
• Poisson's summation formula