Discrete 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

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The discrete Chebyshev polynomial   is a polynomial of degree n in x, for  , constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function   with   being the Dirac delta function. That is,  

The integral on the left is actually a sum because of the delta function, and we have,  

Thus, even though   is a polynomial in  , only its values at a discrete set of points,   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  

Chebyshev chose the normalization so that  

This fixes the polynomials completely along with the sign convention,  .

If the independent variable is linearly scaled and shifted so that the end points assume the values   and  , then as  ,   times a constant, where   is the Legendre polynomial.

Advanced Definition

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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 ≤ km. The task is to approximate f as a polynomial of degree n < m. Consider a positive semi-definite bilinear form   where g and h are continuous on [−1, 1] and let   be a discrete semi-norm. Let   be a family of polynomials orthogonal to each other   whenever i is not equal to k. Assume all the polynomials   have a positive leading coefficient and they are normalized in such a way that  

The   are called discrete Chebyshev (or Gram) polynomials.[3]

Connection with Spin Algebra

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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  , where   is the rotation angle. In other words, if   where   are the usual angular momentum or spin eigenstates, and   then  

The eigenvectors   are scaled and shifted versions of the Chebyshev polynomials. They are shifted so as to have support on the points   instead of   for   with   corresponding to  , and   corresponding to  . In addition, the   can be scaled so as to obey other normalization conditions. For example, one could demand that they satisfy   along with  .

References

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  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, S2CID 116847377
  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 (6): 1447. Bibcode:1958PhRv..111.1447M. doi:10.1103/PhysRev.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 (2): 101. Bibcode:1982FoPh...12..101M. doi:10.1007/BF00736844. S2CID 121648820.
  6. ^ 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.