An approximation of analytical functions by local splines

Abstract

Local splines are presented for the approximation of functions of one and many variables, which are analytic in the domains\(D^\# = \bigcup\limits_{i = l}^l {U_i } \left( {z_i } \right)\), where Ui(zi) is a unit disk in the complex plane Ci,i=1,2,…,l, l=1,2, …. Results are given for functions whose r-order derivatives belong to the Hardy's class Hp,1≤p≤∞. It is shown that the approximation converge to the function at the rate\(A_{e \times p} \left( { - C\sqrt {n\left( {r - 1/p} \right)} } \right)\) for functions of one variable and An−(r−1/p)/(l−1) for functions of l variables, where n is the number of points of local splines and A and C are positive constants.