Discrete Chebyshev polynomials

Not to be confused with 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

They are defined as follows: Let f be a smooth function defined on the closed interval $\left[-1,1\right]$ whose values are known explicitly only at points $\textstyle x_k:=-1+(2k-1)/m$, where k and m are integers and $1\leq k\leq m$. The task is to approximate f as a polynomial of degree n < m. Now 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 $\textstyle\left[-1,1\right]$ and let

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

be a discrete semi-norm. Now let φ_k be a family of polynomials orthogonal to

$\left(g,h\right)_d,$

which have a positive leading coefficient and which 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, http://www.archive.org/stream/oeuvresdepltche01chebrich#page/n551/mode/2up 
• Gram, J. P. (1883), "Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate" (in German), Journal für die reine und angewandte Mathematik 94: 41–73, JFM 15.0321.03, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002158604