In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev^[1] and rediscovered by Gram.^[2] They were later found to be applicable to various algebraic properties of spin angular momentum.

Elementary Definition

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.

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

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

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

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

The ${\displaystyle \varphi _{k}}$ are called discrete Chebyshev (or Gram) polynomials.^[3]

Connection with Spin Algebra

The discrete Chebyshev polynomials have surprising connections to various algebraic properties of spin: spin transition probabilities,^[4] the probabilities for observations of the spin in Bohm's spin-s version of the Einstein-Podolsky-Rosen experiment,^[5] and Wigner functions for various spin states.^[6]

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

$${\displaystyle d_{mm'}=\langle j,m|e^{-i\theta J_{y}}|j,m'\rangle ,}$$

where ${\displaystyle |j,m\rangle }$ are the usual angular momentum or spin eigenstates, and

$${\displaystyle F_{mm'}(\theta )=|d_{mm'}(\theta )|^{2},}$$

then

$${\displaystyle \sum _{m'=-j}^{j}F_{mm'}(\theta )\,f_{\ell }^{j}(m')=P_{\ell }(\cos \theta )f_{\ell }^{j}(m).}$$

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 ${\displaystyle m=-j,-j+1,\ldots ,j}$ instead of ${\displaystyle r=0,1,\ldots ,N}$ 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

$${\displaystyle {\frac {1}{2j+1}}\sum _{m=-j}^{j}f_{\ell }^{j}(m)f_{\ell '}^{j}(m)=\delta _{\ell \ell '},}$$

along with ${\displaystyle f_{\ell }^{j}(j)>0}$.

References