Chebyshev polynomials and best approximation of some classes of functions

Abstract

In this research using properties of Chebyshev polynomialswe explicitly determine the best uniform polynomial approximation of some classes of functions. In this way we present some new theorems about the best approximation of these classes.

Systems of nonlinear Volterra integro-differential equations of arbitrary order

“…Some of Chebyshev polynomials properties are: orthogonality, recursive, real zeros, complete for the space of polynomials, etc. For these reasons, many researchers have employed these polynomials in their research [17,18,19,20,21,22,23].…”

Novel orthogonal functions for solving differential equations of arbitrary order

“…Chebyshev polynomials have many properties, for example orthogonal, recursive, simple real roots, complete in the space of polynomials. For these reasons, many researchers have employed these polynomials in their research [37,38,39,40,41]. Using some transformations, some researchers extended Chebyshev polynomials to semi-infinite or infinite domain, for example by using x = t−L t+L , L > 0 the rational Chebyshev functions on semi-infinite domain [42,43,44,45,46,47,48], by using…”

New Numerical Solution For Solving Nonlinear Singular Thomas-Fermi Differential Equation

“…The Chebyshev polynomials are frequently used in the polynomial approximation, Gauss-quadrature integration, integral and differential equations and Spectral methods, and also have many properties, such as orthogonal, recursive, simple real roots, complete in the space of polynomials. For these reasons, many researchers have used these polynomials in their researches [89,90]. [29] 1930 A Monotone and Taylor's series Bush and Caldwell [30] 1931 N Differential analyzer Sommerfeld [31] 1932 A Asymptotic behavior Feynman et al [34] 1949 A Taylor's series Coulson and March [35] 1949 A Asymptotic series Kobayashi et al [36] 1955 A Asymptotic series Mason [37] 1964 A Rational functions Hille [38] 1970 A Asymptotic behavior and Taylor's series More [39] 1976 A Local density approximation Graef et al [40] 1976 A Asymptotic behavior Laurenzi [41] 1990 A Perturbative procedure MacLeod [42] 1992 N Chebyshev collocation method Al-Zanaidi, Grossmann [43] 1996 N Monotone discretization principle Adomian [44] 1998 A Adomian decomposition method(ADM) Wazwaz [45] 1999 A Pade -ADM Epele et al [46] 1999 A Pade approximant Mandelzweig, Tabakin [47] 2001 N Quasilinearization iteration method Esposito [48] 2002 A Majorana method Kiessling [49] 2002 A Asymptotic behavior Liao [50] 2003 A Homotopy analysis method He [51] 2003 A Hybrid of semi-inverse and Ritz methods Ramos [52] 2004 N Piecewise quasilinearization technique Zaitsev et al [53] 2004 N Iterative and sweep methods Desaix et al [54] 2004 A Direct variational method Khan and Xu [55] 2007 A Homotopy analysis method El-Nahhas [56] 2008 A Homotopy analysis method Iacono [57] 2008 A By exploiting integral properties Yao [58] 2008 A Homotopy analysis method Parand and Shahini [59] 2009 N Rational Chebyshev collocation method Ebaid…”