Mathematics in Computer Science

, Volume 9, Issue 3, pp 365–389 | Cite as

Algorithmic Approach for Formal Fourier Series

  • Wolfram KoepfEmail author
  • Etienne Nana Chiadjeu


The study of trigonometric series has started at the beginning of the nineteenth century. Joseph Fourier made the important observation that almost every function of a closed interval can be decomposed into the sum of sine and cosine functions. This technique to develop a function into a trigonometric series was published for the first time in 1822 by Joseph Fourier. The resulting series is nowadays called Fourier series. Since Fourier’s time, many different approaches to understand the concept of Fourier series have been discovered, each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available at the time Fourier completed his original work. Although the original motivation was to solve the heat equation for a metal plate, it later became obvious that the same technique could be applied to a wide variety of mathematical and physical problems and has many applications in electrical engineering, vibration analysis, acoustics, optics, signal treatment, image processing, etc. Despite the importance of Fourier series, the method used until now to compute them via computer algebra systems (CAS) is essentially based on the same principle as in Fourier’s time, i.e. by the evaluation of certain integrals. Unfortunately this technique is not completely successful for many functions. Although numeric values of the Fourier coefficients might be available, symbolic values are often not accessible. Modern CAS like Maple or Mathematica can compute such integrals in many cases for a given \({n \in \mathbb{Z}}\). However if one is interested in the Fourier coefficients for all \({n \in \mathbb{Z}}\), then n is considered as a given symbolic variable and such integrals can be computed only in few cases. In this paper we introduce an algorithmic approach to compute those Fourier coefficients, involving differential equations of a particular form, and recurrence equations. This approach extrapolates the computation of the Fourier series for functions for which the computation of Fourier coefficients via the definition is out of reach for current CAS.

A holonomic recurrence equation for a n , i.e. a recurrence equation which is linear, homogeneous and has polynomial coefficients, can be written in operator notation as L(a n ) = 0. The operator L can be interpreted as a non-commutative polynomial via the commutator rule NnnN = N, N denoting the shift operator Na n  = a n+1. In the last section we show how our algorithm can be used to factorize such recurrence operators in certain cases.


Fourier series Fourier coefficients Trigonometric holonomic function Holonomic recurrence equation Non-commutative factorization 

Mathematics Subject Classification

33F10 68W30 


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  1. 1.
    Benoit, A.: Algorithmique semi-numerique rapide des series de Tchebychev. Ph.D. Dissertation, cole polytechnique & INRIA, France (2012)Google Scholar
  2. 2.
    Bronstein I., Semendjajew K., Musiol G., Mühlig H.: Taschenbuch der Mathematik. Harri Deutsch, Frankfurt (2008)zbMATHGoogle Scholar
  3. 3.
    Churchill, R., Brown, J.: Fourier Series and Boundary Value Problems, 3rd edn. McGraw-Hill, Maidenherd (1978)Google Scholar
  4. 4.
    Cluzeau T., van Hoeij M.: Computing hypergeometric solutions of linear recurrence equations. Appl. Algebra Eng. Commun. Comput. 17(2), 83–115 (2006)zbMATHCrossRefGoogle Scholar
  5. 5.
    Denkewitz, L.: Fourieranalyse mit Mathematica. Diploma Thesis, HTWK Leipzig, Leipzig (2000).
  6. 6.
    Dirichlet P.: Sur la convergence des series trigonométriques qui servent à représenter une fonction arbitraire entre des limites données. J. Reine Angew. Math. 4(6), 157169 (1829)Google Scholar
  7. 7.
    van Hoeij M.: Factorization of differential operators with rational functions coefficients. J. Symb. Comput. 24, 537–561 (1997)zbMATHCrossRefGoogle Scholar
  8. 8.
    van Hoeij M.: Finite singularities and hypergeometric solutions of linear recurrence equations. J. Pure Appl. Algebra 139, 109–131 (1998)CrossRefGoogle Scholar
  9. 9.
    Horn, P.: Faktorisierung in Schief-Polynomringen. Ph.D. Dissertation. University of Kassel, Kassel (2008).
  10. 10.
    Koepf W.: Power series in computer algebra. J. Symb. Comput. 13(6), 581–603 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Koepf W.: Computeralgebra. In: Eine Algorithmisch Orientierte Einführung. Springer, Berlin (2006)Google Scholar
  12. 12.
    Koepf W.: Hypergeometric summation. An algorithmic approach to summation and special function identities. In: Springer Universitext Series, 2nd edn. Springer, London (2014)Google Scholar
  13. 13.
    Lewanowicz S., Godoy E., Area I., Ronveaux A., Zarzo A.: Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials. Numer. Algorithms 23, 3150 (2000)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Monagan, M.B., Geddes, K.O., Heal, K.M., Labahn, G., Vorkoetter, S.M., McCarron, J., DeMarco, P.: Maple Advanced Programming Guide. Maplesoft, Waterloo (2008)Google Scholar
  15. 15.
    Le Grand Nana Chiadjeu, E.: Algorithmic Computation of Formal Fourier Series. Ph.D. Disseration. University of Kassel, Kassel (2010).
  16. 16.
    Petkovšek M.: Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symb. Comput. 14, 243–264 (1992)zbMATHCrossRefGoogle Scholar
  17. 17.
    Petkovšek, M., Wilf, H.S., Zeilberger, D.: A=B. AK Peters, Ltd., Wellesley (1996)Google Scholar
  18. 18.
    Salvy B., Zimmermann P.: Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Trans. Math. Softw. 20.2, 163–177 (1994)CrossRefGoogle Scholar
  19. 19.
    Stanley R.: Differentiably finite power series. Eur. J. Comb. 1, 175–188 (1980)zbMATHCrossRefGoogle Scholar
  20. 20.
    Stöcker H.: Taschenbuch Mathematischer Formeln und Moderner Verfahren. Harri Deutsch, Frankfurt (1999)zbMATHGoogle Scholar
  21. 21.
    Stuart R.D.: An introduction to Fourier analysis. In: Methuen’s Monographs on Physical Subjects. Methuen & Co. Ltd., London (1961)Google Scholar
  22. 22.
    Werner, W.: Mathematik lernen mit Maple, vol. 2. Dpunkt, Heidelberg (1998)Google Scholar
  23. 23.
    Wolfram S.: The Mathematica Book. Wolfram Media und Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar

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© Springer Basel 2014

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of KasselKasselGermany

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