In this book, a new approach to approximation procedures is developed. This new approach is characterized by the common feature that the procedures are accurate without being convergent as the mesh size tends to zero. This lack of convergence is compensated for by the flexibility in the choice of approximating functions, the simplicity of multi-dimensional generalizations, and the possibility of obtaining explicit formulas for the values of various integral and pseudodifferential operators applied to approximating functions. The developed techniques allow the authors to design new classes of high-order quadrature formulas for integral and pseudodifferential operators, to introduce the concept of approximate wavelets, and to develop new efficient numerical and semi-numerical methods for solving boundary value problems of mathematical physics. The book is intended for researchers interested in approximation theory and numerical methods for partial differential and integral equations.
Quasi-interpolation Error estimates for quasi-interpolation Various basis functions--examples and constructions Approximation of integral operators Cubature of diffraction, elastic, and hydrodynamic potentials Some other cubature problems Approximation by Gaussians Approximate wavelets Cubature over bounded domains More general grids Scattered data approximate approximations Numerical algorithms based upon approximate approximations--linear problems Numerical algorithms based upon approximate approximations--non-linear problems Bibliography Index.