Approximation with Quasi-Splines

Abstract

Introduction. Motivation. Moduli of continuity. Integral and averaging moduli. Spaces and classes of functions. Statement of the extremal problem. Meshes and their characteristics. Survey of the results. 2: Recovery of Functions of One Variable: Optimization theorem. The standard function. The error of the best method in ^IH^Ow. Optimal methods in ^IH^Tw. Recovery of functions of the classes ^IW^Tr^IH^Tw. 3: Recovery of Functions of Several Variables: Best method for recovery of functions of several variables. An optimal method for recovery of functions of several variables. 4: Some Quadrature Formulas: The best quadrature formula for functions of the chass ^IH^Ow. Optimal quadrature formulas for functions of the class ^IH^Ow. An estimation for the error of optimal quadrature formulas for individual functions in ^IH^Ow. Optimization of the quadrature formula of Niederreiter type for the function classes ^IW^OrH^Ow. 5: Optimal Cubature Formulas with restrictions on the Lattice for the Function Classes ^IH^Ow1w2^I(D^T2) and ^IH^Ow^I(D^T2): Best cubature formula. Optimal cubature formulas for the classes ^IH^Ow1w1^I(D^T2) and ^IH^Ow^I(D^T2). 6: Approximation of Functions by Rational Quasi-Splines: Definition of the basic quasi-splines. Examples. Elementary properties of the basic quasi-splines. Convexity of the basic quasi-splines. Some properties of the fundamental rational quasi-splines. Approximation of functions by rational quasi-splines in uniform metrics. Approximation of functions by rational quasi-splines in integral metrics. 7: Applications Representation of the derivatives of the basic quasi-splines. Monotonic sequences of basic quasi-splines. Properties of the solutions of Problem C. non-analyticity of the function ^I^D*l(x). Localization of the zeros of the derivatives of the function ^I^D*l(x). Tabulating integral reoresentation and central moments of the function ^I^D*l(x). References. Author index. Subject index.