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

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.