Discrete Chebyshev polynomials explained

Discrete Chebyshev polynomials should not 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 and rediscovered by Gram. They were later found to be applicable to various algebraic properties of spin angular momentum.

Elementary Definition

The discrete Chebyshev polynomial

t ^{N} _{n(x)} is a polynomial of degree n in x, for n = 0 , 1 , 2 ,\ldots, N -1 , constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function$$w(x)\; =\; \backslash sum\_^\; \backslash delta(x-r),$$with \delta( ⋅ ) being the Dirac delta function. That is,$$\backslash int\_^\; t^N\_n(x)\; t^N\_m\; (x)\; w(x)\; \backslash ,\; dx\; =\; 0\; \backslash quad\; \backslash text\; \backslash quad\; n\; \backslash ne\; m\; .$$

The integral on the left is actually a sum because of the delta function, and we have,$$\backslash sum\_^\; t^N\_n(r)\; t^N\_m\; (r)\; =\; 0\; \backslash quad\; \backslash text\backslash quad\; n\; \backslash ne\; m.$$

Thus, even though

t ^{N} _{n(x)} is a polynomial in x , only its values at a discrete set of points,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$$\backslash sum\_^\; t^N\_n(r)\; t^N\_n\; (s)\; =\; 0\; \backslash quad\; \backslash text\backslash quad\; r\; \backslash ne\; s.$$

Chebyshev chose the normalization so that$$\backslash sum\_^\; t^N\_n(r)\; t^N\_n\; (r)\; =\; \backslash frac\; \backslash prod\_^n\; (N^2\; -\; k^2).$$

This fixes the polynomials completely along with the sign convention,

t ^{N} _{n(N} - 1 ) > 0 .

If the independent variable is linearly scaled and shifted so that the end points assume the values

-1 and 1 , then as N \to inf ty , t ^{N} _{n( ⋅ )} \to P _{n( ⋅ )} times a constant, where P _{n} is the Legendre polynomial.

Advanced Definition

Let be a smooth function defined on the closed interval [&minus;1,&nbsp;1], whose values are known explicitly only at points, where k and m are integers and . The task is to approximate f as a polynomial of degree n < m. Consider a positive semi-definite bilinear form$$\backslash left(g,h\backslash right)\_d:=\backslash frac\backslash sum\_^,$$where and are continuous on [&minus;1,&nbsp;1] and let$$\backslash left\backslash |g\backslash right\backslash |\_d:=(g,g)^\_$$be a discrete semi-norm. Let

\varphi _{k} be a family of polynomials orthogonal to each other$$\backslash left(\backslash varphi\_k,\; \backslash varphi\_i\backslash right)\_d\; =\; 0$$whenever is not equal to . Assume all the polynomials \varphi _{k} have a positive leading coefficient and they are normalized in such a way that$$\backslash left\backslash |\backslash varphi\_k\backslash right\backslash |\_d=1.$$

The

\varphi _{k} are called discrete Chebyshev (or Gram) polynomials.^[1]

Connection with Spin Algebra

The discrete Chebyshev polynomials have surprising connections to various algebraic properties of spin: spin transition probabilities,^[2] the probabilities for observations of the spin in Bohm's spin-s version of the Einstein-Podolsky-Rosen experiment,^[3] and Wigner functions for various spin states.^[4]

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

P _{\ell} (\cos \theta) , where \theta is the rotation angle. In other words, if$$d\_\; =\; \backslash langle\; j,m|e^|j,m\text{'}\backslash rangle,$$where |j,m\rangle are the usual angular momentum or spin eigenstates,and$$F\_(\backslash theta)\; =\; |d\_(\backslash theta)|^2,$$then$$\backslash sum\_^j\; F\_(\backslash theta)\backslash ,\; f^j\_(m\text{'})=\; P\_(\backslash cos\backslash theta)\; f^j\_(m)\; .$$

The eigenvectors

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

References

1. R.W. Barnard . G. Dahlquist . K. Pearce . L. Reichel . K.C. Richards. Gram Polynomials and the Kummer Function. Journal of Approximation Theory. 1998. 94. 128 - 143. 10.1006/jath.1998.3181. free.
2. A. Meckler. Majorana formula. Physical Review. 1958. 111. 6. 1447. 10.1103/PhysRev.111.1447. 1958PhRv..111.1447M.
3. N. D. Mermin . G. M. Schwarz. Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment. Foundations of Physics. 1982. 12. 2. 101. 10.1007/BF00736844. 1982FoPh...12..101M. 121648820.
4. Anupam Garg. The discrete Chebyshev–Meckler–Mermin–Schwarz polynomials and spin algebra. Journal of Mathematical Physics. 2022. 63. 7. 072101. 10.1063/5.0094575. 2022JMP....63g2101G.