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Discrete Chebyshev polynomials

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Title: Discrete Chebyshev polynomials  
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Subject: Pafnuty Chebyshev
Publisher: World Heritage Encyclopedia

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).


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


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


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


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


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


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