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^[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

1. Chebyshev, P. (1864), "Sur l'interpolation", Zapiski Akademii Nauk, 4, Oeuvres Vol 1 p. 539–560
2. Gram, J. P. (1883), "Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate", Journal für die reine und angewandte Mathematik (in German), 1883 (94): 41–73, doi:10.1515/crll.1883.94.41, JFM 15.0321.03
3. R.W. Barnard; G. Dahlquist; K. Pearce; L. Reichel; K.C. Richards (1998). "Gram Polynomials and the Kummer Function". Journal of Approximation Theory. 94: 128–143. doi:10.1006/jath.1998.3181.
4. A. Meckler (1958). "Majorana formula". Physical Review. 111: 1447.
5. N. D. Mermin; G. M. Schwarz (1982). "Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment". Foundations of Physics. 12: 101.
6. Anupam Garg (2022). "The discrete Chebyshev–Meckler–Mermin–Schwarz polynomials and spin algebra". Journal of Mathematical Physics. 63: 072101. doi:10.1063/5.0094575.