Convergence of Spectral Approximations

  • George Rawitscher
  • Victo dos Santos FilhoEmail author
  • Thiago Carvalho Peixoto


In this chapter we discuss by means of general theorems the rate of convergence and accuracy of spectral methods. These are the methods so called “spectral” methods that consist in expanding the solution to a particular problem in terms of a set of basis functions. We initially present theorems about the convergence of Fourier transforms, alongside with the accuracy of a Fourier spectral differentiation. Next, we present theorems concerning the calculation of the interpolation error and the determination of the set of functions that gives rise to the best interpolation in such methods. In last section, we present assignments in order to analyze the rate of convergence and the error of various sets of basis functions in an expansion.


  1. 1.
    L.N. Trefethen, Spectral Methods in MATLAB (SIAM, Philadelphia, 2000)CrossRefGoogle Scholar
  2. 2.
    B.D. Shizgal, Spectral Methods in Chemistry and Physics. Applications to Kinetic Theory and Quantum Mechanics (Springer, Dordrecht, 2015)zbMATHGoogle Scholar
  3. 3.
    A. Deloff, Ann. Phys. (NY) 322, 1373–1419 (2007)Google Scholar
  4. 4.
    S. Širca, M. Horvat, Computational Methods for Physicists. Graduate Texts in Physics (Springer, Berlin, 2012)CrossRefGoogle Scholar
  5. 5.
    Y.L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975)zbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • George Rawitscher
    • 1
  • Victo dos Santos Filho
    • 2
    Email author
  • Thiago Carvalho Peixoto
    • 3
  1. 1.University of ConnecticutStorrsUSA
  2. 2.H4D Scientific Research LaboratoryBela VistaBrazil
  3. 3.Federal Institute of Sergipe (IFS)Nossa Senhora da GlóriaBrazil

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