Discrete Chebyshev polynomials

In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by and rediscovered by .
Elementary Definition.
The discrete Chebyshev polynomial formula_1 is a polynomial of degree "n" in "x",
for formula_2, constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function
with formula_4 being the Dirac delta function. That is,
The integral on the left is actually a sum because of the delta function, and we have,
Thus, even though formula_1 is a polynomial in formula_8, only its values at a discrete set of points,
formula_9 are of any significance. Nevertheless, because these polynomials can be defined in terms of orthogonality with respect to a nonnegative weight function, the entire theory of orthogonal polynomials is applicable. In particular, the polynomials are complete in the sense that
Chebyshev chose the normalization so that
This fixes the polynomials completely along with the sign convention, formula_12.
Advanced Definition.
Let "f" be a smooth function defined on the closed interval [−1, 1], whose values are known explicitly only at points "x""k" := −1 + (2"k" − 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
where "g" and "h" are continuous on [−1, 1] and let
be a discrete semi-norm. Let formula_15 be a family of polynomials orthogonal to each other
whenever i is not equal to k. Assume all the polynomials formula_15 have a positive leading coefficient and they are normalized in such a way that
The formula_15 are called discrete Chebyshev (or Gram) polynomials.