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. This connection shows that they can be expressed as Clebsch-Gordan coefficients.
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) = \sum_{r = 0}^{N-1} \delta(x-r),
with \delta(\cdot) being the Dirac delta function. That is,
\int_{-\infty}^{\infty} t^N_n(x) t^N_m (x) w(x) \, dx = 0 \quad \text{ if } \quad n \ne m .
The integral on the left is actually a sum because of the delta function, and we have,
\sum_{r = 0}^{N-1} t^N_n(r) t^N_m (r) = 0 \quad \text{ if }\quad n \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
\sum_{n = 0}^{N-1} t^N_n(r) t^N_n (s) = 0 \quad \text{ if }\quad r \ne s.
Chebyshev chose the normalization so that
\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, 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 \infty, t^N_n(\cdot) \to P_n(\cdot) times a constant, where 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 xk := −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
\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
\left\|g\right\|_d:=(g,g)^{1/2}_{d}
be a discrete semi-norm. Let \varphi_k be a family of polynomials orthogonal to each other
\left( \varphi_k, \varphi_i\right)_d = 0
whenever i is not equal to k. Assume all the polynomials \varphi_k have a positive leading coefficient and they are normalized in such a way that
\left\|\varphi_k\right\|_d=1.
The \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 P_{\ell}(\cos \theta), where \theta is the rotation angle. In other words, if
d_{mm'} = \langle j,m|e^{-i\theta J_y}|j,m'\rangle,
where |j,m\rangle are the usual angular momentum or spin eigenstates,
and
F_{mm'}(\theta) = |d_{mm'}(\theta)|^2 ,
then
\sum_{m' = -j}^j F_{mm'}(\theta)\, f^j_{\ell}(m')= P_{\ell}(\cos\theta) f^j_{\ell}(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 2j+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
\frac{1}{2j+1} \sum_{m=-j}^{j} f^j_{\ell}(m) f^j_{\ell'}(m) = \delta_{\ell\ell'},
along with f^j_{\ell}(j) > 0.
Connection with Clebsch-Gordan coefficients
In the form, f^j_{\ell}(m), the connection with spin algebra shows that these polynomials are Clebsch-Gordan coefficients. To be consistent with the normalization given above, the relation is
f^j_{\ell}(m) = (-1)^{j - m} \sqrt{2j+1} \langle{j\,j\,\ell\,0|j,m; j,-m}\rangle.
References
- ^ Chebyshev, P. (1864), "Sur l'interpolation", Zapiski Akademii Nauk. 4, Oeuvres Vol 1 p. 539–560
- ^ Gram, J. P. (1883), "Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate" (in German), Journal für die reine und angewandte Mathematik. 1883 (94): 41–73, doi:10.1515/crll.1883.94.41. S2CID 116847377
- ^ 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
- ^ A. Meckler (1958). "Majorana formula". Physical Review. 111 (6): 1447. Bibcode:1958PhRv..111.1447M. doi:10.1103/PhysRev.111.1447
- ^ N. D. Mermin & G. M. Schwarz (1982). "Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment". Foundations of Physics. 12 (2): 101. Bibcode:1982FoPh...12..101M. doi:10.1007/BF00736844. S2CID 121648820
- ^ Anupam Garg (2022). "The discrete Chebyshev–Meckler–Mermin–Schwarz polynomials and spin algebra". Journal of Mathematical Physics. 63 (7): 072101. Bibcode:2022JMP....63g2101G. doi:10.1063/5.0094575