Optimal Cubature Formulae and Recovery of Fast-Oscillatory Functions from an Interpolational Class

Melnik, K.N. and Melnik, R.V.N.

BIT Numerical Mathematics, 41(4), 747--774, 2001


The method of limit functions is used to construct optimal-by-accuracy and optimal-by-order (with constant not exceeding two) cubature formulae for the integration of fast oscillatory functions given by their values at a finite number of fixed nodes in a square region. The construction is based on explicit forms of the majorant and minorant in the given interpolational class $C_{1,L,N}^2$ and the solution of the problem of optimal-by-accuracy recovery of functions from this class. It is shown that an appropriate choice of the grid in this interpolational class leads to a substantial reduction in a priori information required for the application of the proposed approach.