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).
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 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 [−1, 1] and let
be a discrete semi-norm. Let φk be a family of polynomials orthogonal to each other
whenever i is not equal to k. Assume all the polynomials φk have a positive leading coefficient and they are normalized in such a way that
The φk are called discrete Chebyshev (or Gram) polynomials.
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