Discrete Chebyshev polynomials

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 + (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 $${\displaystyle {\Vert g\Vert }_d:=(g,g{)}_d^{1/2}}$$ be a discrete semi-norm. Let ${\displaystyle {\phi }_k}$ 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 ${\displaystyle {\phi }_k}$ have a positive leading coefficient and they are normalized in such a way that $${\displaystyle {\Vert {\phi }_k\Vert }_d=1.}$$

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

Connection with Spin Algebra

The discrete Chebyshev polynomials have surprising connections to various algebraic properties of spin: spin transition probabilities, the probabilities for observations of the spin in Bohm's spin-s version of the Einstein-Podolsky-Rosen experiment, and Wigner functions for various spin states.

Specifically, the polynomials turn out to be the eigenvectors of the absolute square of the rotation matrix (the Wigner D-matrix). The associated eigenvalue is the Legendre polynomial ${\displaystyle P_{\ell }(\cos \theta )}$, where ${\displaystyle \theta }$ is the rotation angle. In other words, if $${\displaystyle d_{m{m}^{\prime }}=⟨j,m|e^{-i\theta J_y}|j,m^{\prime }⟩,}$$ where ${\displaystyle |j,m⟩}$ are the usual angular momentum or spin eigenstates, and $${\displaystyle F_{m{m}^{\prime }}(\theta )=|d_{m{m}^{\prime }}(\theta ){|}^2,}$$ then $${\displaystyle \sum _{m^{\prime }=-j}^j{F}_{m{m}^{\prime }}(\theta )f_{\ell }^j(m^{\prime })=P_{\ell }(\cos \theta )f_{\ell }^j(m).}$$

The eigenvectors ${\displaystyle f_{\ell }^j(m)}$ are scaled and shifted versions of the Chebyshev polynomials. They are shifted so as to have support on the points ${\displaystyle m=-j,-j+1,\dots ,j}$ instead of ${\displaystyle r=0,1,\dots ,N}$ for ${\displaystyle t_n^N(r)}$ with ${\displaystyle N}$ corresponding to ${\displaystyle 2j+1}$, and ${\displaystyle n}$ corresponding to ${\displaystyle \ell }$. In addition, the ${\displaystyle f_{\ell }^j(m)}$ can be scaled so as to obey other normalization conditions. For example, one could demand that they satisfy $${\displaystyle \frac{1}{2j+1}\sum _{m=-j}^j{f}_{\ell }^j(m)f_{{\ell }^{\prime }}^j(m)={\delta }_{\ell {\ell }^{\prime }},}$$ along with ${\displaystyle f_{\ell }^j(j)>0}$.