discrete Chebyshev polynomials

The discrete Chebyshev polynomial $t^N\_n(x)$ is a polynomial of degree n in x, for $n\; =\; 0,\; 1,\; 2,\backslash ldots,\; N\; -1$, constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function

$$w(x)\; =\; \backslash sum\_\{r\; =\; 0\}^\{N-1\}\; \backslash delta(x-r),$$

with $\backslash delta(\backslash cdot)$ being the Dirac delta function. That is,

$$\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; t^N\_n(x)\; t^N\_m\; (x)\; w(x)\; \backslash ,\; dx\; =\; 0\; \backslash quad\; \backslash text\{\; if\; \}\; \backslash quad\; n\; \backslash ne\; m\; .$$

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

$$\backslash sum\_\{r\; =\; 0\}^\{N-1\}\; t^N\_n(r)\; t^N\_m\; (r)\; =\; 0\; \backslash quad\; \backslash text\{\; if\; \}\backslash quad\; n\; \backslash ne\; m.$$

Thus, even though $t^N\_n(x)$ is a polynomial in $x$, only its values at a discrete set of points,

$x\; =\; 0,\; 1,\; 2,\; \backslash ldots,\; N-1$ 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

$$\backslash sum\_\{n\; =\; 0\}^\{N-1\}\; t^N\_n(r)\; t^N\_n\; (s)\; =\; 0\; \backslash quad\; \backslash text\{\; if\; \}\backslash quad\; r\; \backslash ne\; s.$$

Chebyshev chose the normalization so that

$$\backslash sum\_\{r\; =\; 0\}^\{N-1\}\; t^N\_n(r)\; t^N\_n\; (r)\; =\; \backslash frac\{N\}\{2n+1\}\; \backslash prod\_\{k=1\}^n\; (N^2\; -\; k^2).$$

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 $N\; \backslash to\; \backslash infty$, $t^N\_n(\backslash cdot)\; \backslash to\; P\_n(\backslash cdot)$ times a constant, where $P\_n$ is the Legendre polynomial.

Let {{math|f}} be a smooth function defined on the closed interval [−1, 1], whose values are known explicitly only at points {{math|1=x_k := −1 + (2k − 1)/m}}, where k and m are integers and {{math|1 ≤ k ≤ m}}. The task is to approximate f as a polynomial of degree n < m. Consider a positive semi-definite bilinear form

$$\backslash left(g,h\backslash right)\_d:=\backslash frac\{1\}\{m\}\backslash sum\_\{k=1\}^\{m\}\{g(x\_k)h(x\_k)\},$$

where {{math|g}} and {{math|h}} are continuous on [−1, 1] and let

$$\backslash left\backslash |g\backslash right\backslash |\_d:=(g,g)^\{1/2\}\_\{d\}$$

be a discrete semi-norm. Let $\backslash varphi\_k$ be a family of polynomials orthogonal to each other

$$\backslash left(\; \backslash varphi\_k,\; \backslash varphi\_i\backslash right)\_d\; =\; 0$$

whenever {{mvar|i}} is not equal to {{mvar|k}}. Assume all the polynomials $\backslash varphi\_k$ 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 $\backslash varphi\_k$ are called discrete Chebyshev (or Gram) polynomials.{{cite journal

| author = R.W. Barnard |author2=G. Dahlquist |author3=K. Pearce |author4=L. Reichel |author5=K.C. Richards

| title = Gram Polynomials and the Kummer Function

| journal = Journal of Approximation Theory

| year = 1998

| volume = 94

| pages = 128–143

| doi = 10.1006/jath.1998.3181

| doi-access = free

}}

The discrete Chebyshev polynomials have surprising connections to various algebraic properties of spin: spin transition probabilities,{{cite journal

| author = A. Meckler

| title = Majorana formula

| journal = Physical Review

| year = 1958

| volume = 111

| issue = 6

| page = 1447

| doi = 10.1103/PhysRev.111.1447

| bibcode = 1958PhRv..111.1447M

}}

the probabilities for observations of the spin in Bohm's spin-s version of the Einstein-Podolsky-Rosen experiment,{{cite journal

| author = N. D. Mermin | author2 = G. M. Schwarz

| title = Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment

| journal = Foundations of Physics

| year = 1982

| volume = 12

| issue = 2

| page = 101

| doi = 10.1007/BF00736844

| bibcode = 1982FoPh...12..101M

| s2cid = 121648820

}}

and Wigner functions for various spin states.{{cite journal

| author = Anupam Garg

| title = The discrete Chebyshev–Meckler–Mermin–Schwarz polynomials and spin algebra

| journal = Journal of Mathematical Physics

| year = 2022

| volume = 63

| issue = 7

| page = 072101

| doi = 10.1063/5.0094575

| bibcode = 2022JMP....63g2101G

}}

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 $P\_\{\backslash ell\}(\backslash cos\; \backslash theta)$, where $\backslash theta$ is the rotation angle. In other words, if

$$d\_\{mm\text{'}\}\; =\; \backslash langle\; j,m|e^\{-i\backslash theta\; J\_y\}|j,m\text{'}\backslash rangle,$$

where $|j,m\backslash rangle$ are the usual angular momentum or spin eigenstates,

and

$$F\_\{mm\text{'}\}(\backslash theta)\; =\; |d\_\{mm\text{'}\}(\backslash theta)|^2\; ,$$

then

$$\backslash sum\_\{m\text{'}\; =\; -j\}^j\; F\_\{mm\text{'}\}(\backslash theta)\backslash ,\; f^j\_\{\backslash ell\}(m\text{'})=\; P\_\{\backslash ell\}(\backslash cos\backslash theta)\; f^j\_\{\backslash ell\}(m)\; .$$

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

$$\backslash frac\{1\}\{2j+1\}\; \backslash sum\_\{m=-j\}^\{j\}\; f^j\_\{\backslash ell\}(m)\; f^j\_\{\backslash ell\text{'}\}(m)\; =\; \backslash delta\_\{\backslash ell\backslash ell\text{'}\},$$

along with $f^j\_\{\backslash ell\}(j)\; >\; 0$.

Category:Orthogonal polynomials

Category:Approximation theory