Discrete Chebyshev polynomials

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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[edit]

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

\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 φk 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 φk have a positive leading coefficient and they are normalized in such a way that

\left\|\phi_k\right\|_d=1.

The φk are called discrete Chebyshev (or Gram) polynomials.[1]

References[edit]

  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.