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). 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). Let f be a smooth function defined on the closed interval , whose values are known explicitly only at points xk := −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 where g and h are continuous on and let be a discrete semi-norm. Let φ k {displaystyle varphi _{k}} be a family of polynomials orthogonal to each other whenever i is not equal to k. Assume all the polynomials φ k {displaystyle varphi _{k}} have a positive leading coefficient and they are normalized in such a way that The φ k {displaystyle varphi _{k}} are called discrete Chebyshev (or Gram) polynomials.