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.

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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_{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,

\sum _{r=0}^{N-1}t_{n}^{N}(r)t_{m}^{N}(r)=0\quad {\text{ if }}\quad n\neq 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\neq 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 $x k := -1 + (2 k - 1)/ m$ , where k and m are integers and $1 \leq k \leq m$ . 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)_{d}^{1/2}

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_{\ell }^{j}(m')=P_{\ell }(\cos \theta )f_{\ell }^{j}(m).

The eigenvectors $f_{\ell }^{j}(m)$ 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 $f_{\ell }^{j}(m)$ 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_{\ell }^{j}(m)f_{\ell '}^{j}(m)=\delta _{\ell \ell '},

along with

f_{\ell }^{j}(j)>0

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", 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
^ 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.

This page was last edited on 13 December 2023, at 01:28

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