Discrete Chebyshev polynomials
In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev (1864) and rediscovered by Gram (1883).
The polynomials are defined as follows: Let f be a smooth function defined on the closed interval [−1, 1], whose values are known explicitly only at points xk := −1 + (2k − 1)/m, where k and m are integers and 1 ≤ k ≤ m. The task is to approximate f as a polynomial of degree n < m. Consider a positive semi-definite bilinear form
where g and h are continuous on [−1, 1] and let
The φk are called discrete Chebyshev (or Gram) polynomials.
- R.W. Barnard; G. Dahlquist, K. Pearce, L. Reichel, K.C. Richards (1998). "Gram Polynomials and the Kummer Function". Journal of Approximation Theory 94: 128–143. doi:10.1006/jath.1998.3181.
- Chebyshev, P. (1864), "Sur l'interpolation", Zapiski Akademii Nauk 4, Oeuvres Vol 1 p. 539–560
- Gram, J. P. (1883), "Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate", Journal für die reine und angewandte Mathematik (in German) 94: 41–73, doi:10.1515/crll.1883.94.41, JFM 15.0321.03