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).

Definition

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 x_k := −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

$\left(g,h\right)_d:=\frac{1}{m}\sum_{k=1}^{m}{g(x_k)h(x_k)},$

where g and h are continuous on [−1, 1] and let

$\left\|g\right\|_d:=(g,g)^{1/2}_{d}$

be a discrete semi-norm. Let φ_k be a family of polynomials orthogonal to each other

$\left( \phi_k, \phi_i\right)_d=0$

whenever i is not equal to k. Assume all the polynomials φ_k have a positive leading coefficient and they are normalized in such a way that

$\left\|\phi_k\right\|_d=1.$

The φ_k are called discrete Chebyshev (or Gram) polynomials.^[1]

References

1. 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