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# Discrete Chebyshev polynomials

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 Title: Discrete Chebyshev polynomials Author: World Heritage Encyclopedia Language: English Subject: Pafnuty Chebyshev Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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

They 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. Now consider a positive semi-definite bilinear form

$\left\left(g,h\right\right)_d:=\frac\left\{1\right\}\left\{m\right\}\sum_\left\{k=1\right\}^\left\{m\right\}\left\{g\left(x_k\right)h\left(x_k\right)\right\},$

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

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

be a discrete semi-norm. Now let φk be a family of polynomials orthogonal to

$\left\left(g,h\right\right)_d,$

which have a positive leading coefficient and which are normalized in such a way that

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

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

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