In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gram. They were later found to be applicable to various algebraic properties of spin angular momentum.
Elementary Definition.
The discrete Chebyshev polynomial formula_1 is a polynomial of degree "n" in "x", for formula_2, constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function
formula_3
with formula_4 being the Dirac delta function. That is,
formula_5
The integral on the left is actually a sum because of the delta function, and we have,
formula_6
Thus, even though formula_1 is a polynomial in formula_8, only its values at a discrete set of points,
formula_9 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
formula_10
Chebyshev chose the normalization so that
formula_11
This fixes the polynomials completely along with the sign convention, formula_12.
If the independent variable is linearly scaled and shifted so that the end points assume the values formula_13 and formula_14, then as formula_15, formula_16 times a constant, where formula_17 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 ≤ "k" ≤ "m". The task is to approximate "f" as a polynomial of degree "n" ; "m". Consider a positive semi-definite bilinear form
formula_18
where "g" and "h" are continuous on [−1, 1] and let
formula_19
be a discrete semi-norm. Let formula_20 be a family of polynomials orthogonal to each other
formula_21
whenever i is not equal to k. Assume all the polynomials formula_20 have a positive leading coefficient and they are normalized in such a way that
formula_23
The formula_20 are called discrete Chebyshev (or Gram) polynomials.
Connection with Spin Algebra.
The discrete Chebyshev polynomials have surprising connections to various algebraic properties of spin: spin transition probabilities,
the probabilities for observations of the spin in Bohm's spin-s version of the Einstein-Podolsky-Rosen experiment,
and Wigner functions for various spin states.
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 formula_25, where formula_26 is the rotation angle. In other words, if
formula_27
where formula_28 are the usual angular momentum or spin eigenstates,
and
formula_29
then
formula_30
The eigenvectors formula_31 are scaled and shifted versions of the Chebyshev polynomials. They are shifted so as to have support on the points formula_32 instead of formula_33 for formula_34 with formula_35 corresponding to formula_36, and formula_37 corresponding to formula_38. In addition, the formula_31 can be scaled so as to obey other normalization conditions. For example, one could demand that they satisfy
formula_40
along with formula_41.