The Fourier Transform

  • Kenneth Lange
Part of the Statistics and Computing book series (SCO)


The Fourier transform is one of the most productive tools of the mathematical sciences. It crops up again and again in unexpected applications to fields as diverse as differential equations, numerical analysis, probability theory, number theory, quantum mechanics, optics, medical imaging, and signal processing [3, 5, 7, 8, 9]. One explanation for its wide utility is that it turns complex mathematical operations like differentiation and convolution into simple operations like multiplication. Readers most likely are familiar with the paradigm of transforming a mathematical equation, solving it in transform space, and then inverting the solution. Besides its operational advantages, the Fourier transform often has the illuminating physical interpretation of decomposing a temporal process into component processes with different frequencies.


Fourier Transform Saddle Point Integrable Function Moment Generate Function Saddle Point Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barndorff-Nielsen OE, Cox DR (1989) Asymptotic Techniques for Use in Statistics. Chapman and Hall, LondonMATHGoogle Scholar
  2. 2.
    Daniels HE (1982) The saddlepoint approximation for a general birth process. J Appl Prob 19:20-28CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Dym H, McKean HP (1972) Fourier Series and Integrals. Academic Press, New YorkMATHGoogle Scholar
  4. 4.
    Feller W (1971) An Introduction to Probability Theory and Its Applications, Volume 2, 2nd ed. Wiley, New YorkGoogle Scholar
  5. 5.
    Folland GB (1992) Fourier Analysis and its Applications. Wadsworth and Brooks/Cole, Pacific Grove, CAMATHGoogle Scholar
  6. 6.
    Hall P (1992) The Bootstrap and Edgeworth Expansion. Springer, New YorkGoogle Scholar
  7. 7.
    Körner TW (1988) Fourier Analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  8. 8.
    Lighthill MJ (1958) An Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press, CambridgeGoogle Scholar
  9. 9.
    Rudin W (1973) Functional Analysis. McGraw-Hill, New YorkMATHGoogle Scholar

Copyright information

© Springer New York 2010

Authors and Affiliations

  1. 1.Departments of Biomathematics, Human Genetics, and Statistics David Geffen School of MedicineUniversity of California, Los AngelesLos AngelesUSA

Personalised recommendations