In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gram. They were later found to be applicable to various algebraic properties of spin angular momentum.

The discrete Chebyshev polynomial ${\displaystyle t_{n}^{N}(x)}$ is a polynomial of degree n in x, for ${\displaystyle n=0,1,2,\ldots ,N-1}$, constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function $${\displaystyle w(x)=\sum _{r=0}^{N-1}\delta (x-r),}$$ with ${\displaystyle \delta (\cdot )}$ being the Dirac delta function. That is, $${\displaystyle \int _{-\infty }^{\infty }t_{n}^{N}(x)t_{m}^{N}(x)w(x)\,dx=0\quad {\text{ if }}\quad n\neq m.}$$

The integral on the left is actually a sum because of the delta function, and we have, $${\displaystyle \sum _{r=0}^{N-1}t_{n}^{N}(r)t_{m}^{N}(r)=0\quad {\text{ if }}\quad n\neq m.}$$

Thus, even though ${\displaystyle t_{n}^{N}(x)}$ is a polynomial in ${\displaystyle x}$, only its values at a discrete set of points, ${\displaystyle x=0,1,2,\ldots ,N-1}$ 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 $${\displaystyle \sum _{n=0}^{N-1}t_{n}^{N}(r)t_{n}^{N}(s)=0\quad {\text{ if }}\quad r\neq s.}$$

Chebyshev chose the normalization so that $${\displaystyle \sum _{r=0}^{N-1}t_{n}^{N}(r)t_{n}^{N}(r)={\frac {N}{2n+1}}\prod _{k=1}^{n}(N^{2}-k^{2}).}$$

This fixes the polynomials completely along with the sign convention, ${\displaystyle t_{n}^{N}(N-1)>0}$.

If the independent variable is linearly scaled and shifted so that the end points assume the values ${\displaystyle -1}$ and ${\displaystyle 1}$, then as ${\displaystyle N\to \infty }$, ${\displaystyle t_{n}^{N}(\cdot )\to P_{n}(\cdot )}$ times a constant, where ${\displaystyle P_{n}}$ is the Legendre polynomial.

Let f be a smooth function defined on the closed interval [−1, 1], whose values are known explicitly only at points x_k := −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 ${\displaystyle \varphi _{k}}$ be a family of polynomials orthogonal to each other $${\displaystyle \left(\varphi _{k},\varphi _{i}\right)_{d}=0}$$ whenever i is not equal to k. Assume all the polynomials ${\displaystyle \varphi _{k}}$ have a positive leading coefficient and they are normalized in such a way that $${\displaystyle \left\|\varphi _{k}\right\|_{d}=1.}$$