Chebyshev Polynomials

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Chebyshev polynomials

[′cheb·ə·shəf ‚päl·i′nō·mē·əlz]
(mathematics)
A family of orthogonal polynomials which solve Chebyshev's differential equation.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Chebyshev Polynomials

 

Chebyshev polynomials of the first kind are a special system of polynomials of successively increasing degree. For n = 0, 1, 2, . . . they are defined by the formula

In particular, T0 = 1, T1 = x, T2 = 2x2 – 1, T3 = 4x3 – 3x and T4 = 8x4 – 8x2 + 1.

The polynomials Tn (x) are orthogonal with respect to the weight function (1 – x2)–½ on the interval [–1, +1] (seeORTHOGONAL POLYNOMIALS). They satisfy the differential equation

(1 – x2)yxy + n2y = 0

and the recursion formula

Tn+1 (x) = 2xTn(x) – Tn – 1(x)

Chebyshev polynomials of the first kind are a special case of the Jacobi polynomials Pn(α,β)(x):

Chebyshev polynomials of the second kind Un (x) are a system of polynomials that are orthogonal with respect to the weight function (1 – x2)½ on the interval [–1, +1]. The relation between Chebyshev polynomials of the second kind and Chebyshev polynomials of the first kind is given by, for example, the recursion formula

(1 – x2)Un – 1 (x) = xTn (x) – Tn+l(x)

REFERENCES

Chebyshev, P. L. Poln. sobr. soch., vols. 2–3. Moscow-Leningrad, 1947–48.
Szegö, G. Ortogonal’nye mnogochleny. Moscow, 1962. (Translated from English.)
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
where [U.sub.n-1] is the Chebyshev polynomial of the second kind of degree n - 1.
where [T.sub.j] - is a Chebyshev polynomial of the first kind with a degree j [39].
Now, [t.sup.k-[alpha]] can be expressed approximately in terms of shifted pseudo-spectral Chebyshev polynomial series, so we have
The enhanced Chebyshev polynomial [T.sub.n](x) is a polynomial in x of degree n and is defined by the following relation:
The Chebyshev polynomial [18] [T.sub.n](x) : [-1,1] [right arrow] [-1,1] is defined as [T.sub.n](x) = cos(n co[s.sup.-1](x)).
Term [T.sub.k] (x) in the above expression denotes a k degree Chebyshev polynomial of the first kind.
But here we use the Chebyshev polynomial of the first kind to replace the monomial term [s.sup.n], instead of the Chebyshev polynomial of the second kind as shown in [16].
In the following, various phase functions are expanded by Legendre polynomial and the second kind of Chebyshev polynomial. In Section 3, the accuracies of the scattering phase functions reconstructed by Chebyshev and Legendre polynomial expansions are discussed.
Elipe, "Integration of orbital motions with chebyshev polynomial," in Dynamics and Astrometry of Natural and Artificial Celestial Bodies, pp.
For example, the Chebyshev polynomial approximation [18-20], one form of orthogonal polynomial approximation, is applied to explore dynamical behaviors in the integer-order systems with bounded random parameters, for example, period-doubling bifurcation, symmetry-breaking bifurcation, Hopf bifurcation, and chaos.
Mason, "Chebyshev polynomial approximations for the L-membrane eigenvalue problem," SIAM Journal on Applied Mathematics, vol.
is the kth Fourier-Chebyshev coefficient and [T.sub.k] (u) = cos (k arccos u) is the kth Chebyshev polynomial. For this case a suitable cosine operator function (see [1,3]) is