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

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

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

be a discrete semi-norm. Let $\varphi _{k}$ be a family of polynomials orthogonal to each other

$\left(\varphi _{k},\varphi _{i}\right)_{d}=0$

whenever i is not equal to k. Assume all the polynomials $\varphi _{k}$ have a positive leading coefficient and they are normalized in such a way that

$\left\|\varphi _{k}\right\|_{d}=1.$

The $\varphi _{k}$ are called discrete Chebyshev (or Gram) polynomials.

References

• 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