An approximation of analytical functions by local splines

Local splines are presented for the approximation of functions of one and many variables, which are analytic in the domains , where U_i(z_i) is a unit disk in the complex plane C_i,i=1,2,…,l, l=1,2, …. Results are given for functions whose r-order derivatives belong to the Hardy's class H_p,1≤p≤∞. It is shown that the approximation converge to the function at the rate 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. This work was supported by Russian Foundation of Fundumental Inverstigations