Orthogonal Polynomials

  • Vilmos Komornik
Part of the Springer Undergraduate Mathematics Series book series (SUMS)


The Chebyshev polynomials have an interesting orthogonality property. Namely, using the orthogonality of the trigonometrical functions cosnt on (0, π), we obtain by the change of variable t = arccosx that T n (x) = cosnt, dtdx = −(1 − x2)−1∕2, and therefore \(\int _{-1}^{1}T_{n}(x)T_{k}(x)(1 - x^{2})^{-1/2}\ dx =\int _{ 0}^{\pi }\cos nt\cos kt\ dt = 0\) for all nk.


  1. 114.
    R. Courant, D. Hilbert, Methods of Mathematical Physics I, Wiley, New York, 1953.Google Scholar
  2. 254.
    D. Jackson, Fourier Series and Orthogonal Polynomials, Menasha, Wisconsin, 1941.Google Scholar
  3. 359.
    I.P. Natanson, Constructive Function Theory I-III, Frederick Ungar, New York, 1961–1965.Google Scholar
  4. 476.
    G. Szegő, Orthogonal Polynomials, American Math. Soc., Providence, Rhode Island, 1975.Google Scholar

Copyright information

© Springer-Verlag London Ltd. 2017

Authors and Affiliations

  • Vilmos Komornik
    • 1
  1. 1.Department of MathematicsUniversity of StrasbourgStrasbourgFrance

Personalised recommendations