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.
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
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
^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
^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.
^N. D. Mermin; G. M. Schwarz (1982). "Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment". Foundations of Physics. 12: 101.
^Anupam Garg (2022). "The discrete Chebyshev–Meckler–Mermin–Schwarz polynomials and spin algebra". Journal of Mathematical Physics. 63: 072101. doi:10.1063/5.0094575.
February 18, 2023
discrete, chebyshev, polynomials, confused, with, chebyshev, polynomials, mathematics, discrete, chebyshev, polynomials, gram, polynomials, type, discrete, orthogonal, polynomials, used, approximation, theory, introduced, pafnuty, chebyshev, rediscovered, gram. Not to be confused with 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 Contents 1 Elementary Definition 2 Advanced Definition 3 Connection with Spin Algebra 4 ReferencesElementary Definition EditThe 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 functionw x r 0 N 1 d x r displaystyle w x sum r 0 N 1 delta x r with d 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 gt 0 displaystyle t n N N 1 gt 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 EditLet 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 lt 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 f k displaystyle varphi k be a family of polynomials orthogonal to each other f k f i d 0 displaystyle left varphi k varphi i right d 0 whenever i is not equal to k Assume all the polynomials f k displaystyle varphi k have a positive leading coefficient and they are normalized in such a way that f k d 1 displaystyle left varphi k right d 1 The f k displaystyle varphi k are called discrete Chebyshev or Gram polynomials 3 Connection with Spin Algebra EditThe 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 8 displaystyle P ell cos theta where 8 displaystyle theta is the rotation angle In other words ifd m m j m e i 8 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 8 d m m 8 2 displaystyle F mm theta d mm theta 2 then m j j F m m 8 f ℓ j m P ℓ cos 8 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 satisfy1 2 j 1 m j j f ℓ j m f ℓ j m d 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 gt 0 displaystyle f ell j j gt 0 References Edit 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 fur die reine und angewandte Mathematik in German 1883 94 41 73 doi 10 1515 crll 1883 94 41 JFM 15 0321 03 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 1447 N D Mermin G M Schwarz 1982 Joint distributions and local realism in the higher spin Einstein Podolsky Rosen experiment Foundations of Physics 12 101 Anupam Garg 2022 The discrete Chebyshev Meckler Mermin Schwarz polynomials and spin algebra Journal of Mathematical Physics 63 072101 doi 10 1063 5 0094575 Retrieved from https en wikipedia org w index php title Discrete Chebyshev polynomials amp oldid 1129771727, wikipedia, wiki, book, books, library,