Some Estimates to Best Approximation of Functions by Fourier-Jacobi Differential Operators

Abstract

In this paper, an extension of the idea of the best approximation in the Hölder spaces with respect to Fourier-Jacobi operators by moduli of smoothness is studied. A special form of the moduli of smoothness is considered to get a strong convergence. Further, advanced approaches of approximation and some direct and inverse results are proved. Moreover, the Jackson-type estimate of functions in Hölder spaces by Jacobi transformations to algebraic polynomials with generalized de la Vallée Poussin mean are established.

The Approximation of Weighted Hölder Functions by Fourier-Jacobi Polynomials to the Singular Sturm-Liouville Operator

“…Properties for moduli of smoothness and approximation's degree of functions in weighted Hölder spaces have significant applications for general theory of partial differential equations in non-smoothness domains and estimates of asymptotic [9][10] . Also, numerical applications for modeling of fractional differential equations in applied sciences and engineering to the natural way have been considered in important studies [11][12][13][14][15][16][17] . This paper aims to improve the previous results in 18 and that get the functions in the Holder spaces which have best approximation by Jacobi differential operator to a 2 nd -order singular Sturm-Liouville type equation.…”