Boyd, "Trouble with GEGenbauer reconstruction for defeating GIBbs' phenomenon: Runge phenomenon in the diagonal limit of

GEGenbauer polynomial approximations," Journal of Computational Physics, vol.

Meng, "A-PDF and

gegenbauer polynomial approximation for dynamic response problems of random structures," Acta Mechanica Sinica, vol.

In 2003, the evolutionary random response problems of the linear systems with random parameters were firstly studied [20, 21] by Chebyshev polynomial and

Gegenbauer polynomial. The period-doubling bifurcation in double-well stochastic Duffing system was analyzed in [22].

where [C.sup.n/2.sub.k] (t) is a

Gegenbauer polynomial ([7]).

They re-express the function in a non-periodic basis set, such as the

Gegenbauer polynomial basis set [6, 7], the Pade Fourier rational basis function [8] and the Freund polynomial basis set [9], to obtain the exponential convergence over the entire interval of the function, including the discontinuities or boundaries.

The last equality allows to express the derivative of a

Gegenbauer polynomial through a sum of

Gegenbauer polynomials of smaller degree.

where [C.sup.[lambda].sub.n] is the

Gegenbauer polynomial of degree n and [V.sub.k] is the intertwining operator, which is a linear operator uniquely determined by [V.sub.k]1 = 1 and [D.sub.j] [V.sub.k] = [V.sub.k] [[partial derivative].sub.j]1 [less than or equal to] j [less than or equal to] d + 1.

One of the formulas is given in [5, Theorem 1.1] and expressed as an infinite sum of which each summand consists of the modified Bessel functions, the

Gegenbauer polynomials and the densities [p.sub.r.sup(0)] (.;x).

The Jacoby polynomials are generalized orthonormal polynomials containing some orthonormal polynomials such as Legendre, Chebyshev, and

Gegenbauer polynomials. For example, the choice [alpha] = [beta] = -1/2 yields the Chebyshev polynomials of the first kind, while choosing [alpha] = [beta] = 1/2 gives the Chebyshev polynomials of the second kind.

Author Brian George Spencer Doman examines classical orthogonal polynomials and their additional properties, covering hermite polynomials, associated Laguerre polynomials, Legendre polynomials, Chebyshev polynomials,

Gegenbauer polynomials, associated Legendre functions, Jacobi polynomials, and many other related mathematical subjects over twelve chapters and appendices.

Gegenbauer polynomials or ultraspherical polynomials [C.sup.[alpha].sub.n](x) are orthogonal polynomials on the interval [-1,1] with respect to the weight function [(1-[x.sup.2]).sup.[alpha]-1/2].

This can be done for the unispherical windows [3] based on

Gegenbauer polynomials as well as for windows proposed by Zierhofer [4].