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 Jørgen Pedersen Gram in 1883.

Definition

They are defined as follows: Let f be a smooth function defined on the closed interval $\left[-1,1\right]$ whose values are known explicitly only at points $\textstyle x_{k}:=-1+(2k-1)/m$, where k and m are integers and $1\leq k\leq m$. The task is to approximate f as a polynomial of degree n < m. Now 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 $\textstyle \left[-1,1\right]$ and let

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

be a discrete semi-norm. Now let $\phi _{k}$ be a family of polynomials orthogonal to

$\left(g,h\right)_{d},$

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

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

The $\phi _{k}$ are called discrete Chebyshev (or Gram) polynomials.

References

• Chebyshev, P. (1864), "Sur l'interpolation", Zapiski Akademii Nauk, 4, Oeuvres Vol 1 p. 539–560 {{citation}}: Cite has empty unknown parameter: |1= (help)