# Discrete Chebyshev polynomials

(Redirected from Gram polynomial)

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

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

${\displaystyle \left(g,h\right)_{d}:={\frac {1}{m}}\sum _{k=1}^{m}{g(x_{k})h(x_{k})},}$

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

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

be a discrete semi-norm. Let φk be a family of polynomials orthogonal to each other

${\displaystyle \left(\phi _{k},\phi _{i}\right)_{d}=0}$

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

${\displaystyle \left\|\phi _{k}\right\|_{d}=1.}$

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

## References

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.