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).

Elementary Definition

The discrete Chebyshev polynomial ${\displaystyle t_{n}^{N}(x)}$ is a polynomial of degree n in x, for ${\displaystyle n=0,1,2,\ldots ,N-1}$, constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function

${\displaystyle w(x)=\sum _{r=0}^{N-1}\delta (x-r)}$

with ${\displaystyle \delta (\cdot )}$ being the Dirac delta function. That is,

${\displaystyle \int _{-\infty }^{\infty }t_{n}^{N}(x)t_{m}^{N}(x)w(x)=0\quad {\rm {if\ }}\quad n\neq m.}$

The integral on the left is actually a sum because of the delta function, and we have,

${\displaystyle \sum _{r=0}^{N-1}t_{n}^{N}(r)t_{m}^{N}(r)=0\quad {\rm {if\ }}\quad n\neq m.}$

Thus, even though ${\displaystyle t_{n}^{N}(x)}$ is a polynomial in ${\displaystyle x}$, only its values at a discrete set of points, ${\displaystyle x=0,1,2,\ldots ,N-1}$ are of any significance. Nevertheless, because these polynomials can be defined in terms of orthogonality with respect to a nonnegative weight function, the entire theory of orthogonal polynomials is applicable. In particular, the polynomials are complete in the sense that

${\displaystyle \sum _{n=0}^{N-1}t_{n}^{N}(r)t_{n}^{N}(s)=0\quad {\rm {if\ }}\quad r\neq s.}$

Chebyshev chose the normalization so that

${\displaystyle \sum _{r=0}^{N-1}t_{n}^{N}(r)t_{n}^{N}(r)={\frac {N}{2n+1}}\prod _{k=1}^{n}(N^{2}-k^{2}).}$

This fixes the polynomials completely along with the sign convention, ${\displaystyle t_{n}^{N}(N-1)>0}$.

Advanced 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

${\displaystyle \left(g,h\right)_{d}:={\frac {1}{m}}\sum _{k=1}^{m}{g(x_{k})h(x_{k})},}$

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

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

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

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

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

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

The ${\displaystyle \varphi _{k}}$ 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), 1883 (94): 41–73, doi:10.1515/crll.1883.94.41, JFM 15.0321.03