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

The discrete Chebyshev polynomial t n N ( x ) {\displaystyle t_{n}^{N}(x)} is a polynomial of degree n in x, for n = 0 , 1 , 2 , … , N − 1 {\displaystyle 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 ) = ∑ r = 0 N − 1 δ ( x − r ) , {\displaystyle w(x)=\sum _{r=0}^{N-1}\delta (x-r),}
with δ ( ⋅ ) {\displaystyle \delta (\cdot )} being the Dirac delta function. That is,
∫ − ∞ ∞ t n N ( x ) t m N ( x ) w ( x ) d x = 0  if  n ≠ m . {\displaystyle \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,

∑ r = 0 N − 1 t n N ( r ) t m N ( r ) = 0  if  n ≠ m . {\displaystyle \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 ) {\displaystyle t_{n}^{N}(x)} is a polynomial in x {\displaystyle x} , only its values at a discrete set of points, x = 0 , 1 , 2 , … , N − 1 {\displaystyle 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

∑ n = 0 N − 1 t n N ( r ) t n N ( s ) = 0  if  r ≠ s . {\displaystyle \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

∑ r = 0 N − 1 t n N ( r ) t n N ( r ) = N 2 n + 1 ∏ k = 1 n ( N 2 − k 2 ) . {\displaystyle \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 {\displaystyle t_{n}^{N}(N-1)>0} .

If the independent variable is linearly scaled and shifted so that the end points assume the values − 1 {\displaystyle -1} and 1 {\displaystyle 1} , then as N → ∞ {\displaystyle N\to \infty } , t n N ( ⋅ ) → P n ( ⋅ ) {\displaystyle t_{n}^{N}(\cdot )\to P_{n}(\cdot )} times a constant, where P n {\displaystyle 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

( g , h ) d := 1 m ∑ k = 1 m g ( x k ) h ( x k ) , {\displaystyle \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
‖ g ‖ d := ( g , g ) d 1 / 2 {\displaystyle \left\|g\right\|_{d}:=(g,g)_{d}^{1/2}}
be a discrete semi-norm. Let φ k {\displaystyle \varphi _{k}} be a family of polynomials orthogonal to each other
( φ k , φ i ) d = 0 {\displaystyle \left(\varphi _{k},\varphi _{i}\right)_{d}=0}
whenever i is not equal to k. Assume all the polynomials φ k {\displaystyle \varphi _{k}} have a positive leading coefficient and they are normalized in such a way that
‖ φ k ‖ d = 1. {\displaystyle \left\|\varphi _{k}\right\|_{d}=1.}

The φ k {\displaystyle \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 ℓ ( cos ⁡ θ ) {\displaystyle P_{\ell }(\cos \theta )} , where θ {\displaystyle \theta } is the rotation angle. In other words, if

d m m ′ = ⟨ j , m | e − i θ J y | j , m ′ ⟩ , {\displaystyle d_{mm'}=\langle j,m|e^{-i\theta J_{y}}|j,m'\rangle ,}
where | j , m ⟩ {\displaystyle |j,m\rangle } are the usual angular momentum or spin eigenstates, and
F m m ′ ( θ ) = | d m m ′ ( θ ) | 2 , {\displaystyle F_{mm'}(\theta )=|d_{mm'}(\theta )|^{2},}
then
∑ m ′ = − j j F m m ′ ( θ ) f ℓ j ( m ′ ) = P ℓ ( cos ⁡ θ ) f ℓ j ( m ) . {\displaystyle \sum _{m'=-j}^{j}F_{mm'}(\theta )\,f_{\ell }^{j}(m')=P_{\ell }(\cos \theta )f_{\ell }^{j}(m).}

The eigenvectors f ℓ j ( m ) {\displaystyle 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 , … , j {\displaystyle m=-j,-j+1,\ldots ,j} instead of r = 0 , 1 , … , N {\displaystyle r=0,1,\ldots ,N} for t n N ( r ) {\displaystyle t_{n}^{N}(r)} with N {\displaystyle N} corresponding to 2 j + 1 {\displaystyle 2j+1} , and n {\displaystyle n} corresponding to ℓ {\displaystyle \ell } . In addition, the f ℓ j ( m ) {\displaystyle f_{\ell }^{j}(m)} can be scaled so as to obey other normalization conditions. For example, one could demand that they satisfy

1 2 j + 1 ∑ m = − j j f ℓ j ( m ) f ℓ ′ j ( m ) = δ ℓ ℓ ′ , {\displaystyle {\frac {1}{2j+1}}\sum _{m=-j}^{j}f_{\ell }^{j}(m)f_{\ell '}^{j}(m)=\delta _{\ell \ell '},}
along with f ℓ j ( j ) > 0 {\displaystyle f_{\ell }^{j}(j)>0} .

References

  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.