This paper continues the theme of the recent work [3] and further develops the Petrov-Galerkin method for Fredholm integral equations of the second kind. Specifically, we study wavelet Petrov-Galerkin schemes based on discontinuous orthogonal multiwavelets and prove that the condition number of the coefficient matrix for the linear system obtained from the wavelet Petrov–Galerkin scheme is bounded. In addition, we propose a truncation strategy which forms a basis for fast wavelet algorithms and analyze the order of convergence and computational complexity of these algorithms.

The study of operator equations is an important branch of mathematics. The fundamental theory of operator equations, linear or nonlinear, formulated in a Hilbert or a Banach space setting, is originated from the classical theory of differential and integral equations. This modern theory of differential and integral equations has been well developed in the last few decades, in which many profound concepts, results, methods, and algorithms were established with considerable generality. On the one hand, various types of mathematical equations, such as linear and nonlinear differential, integral, integro-differential, and functional equations, can all be unified under the same framework of abstract operator equations. On the other hand, many well-known theories and methods in Functional Analysis and Operator Theory have proven very effective and useful in the study of basic solvability problems in operator equations, including not only the existence and uniqueness of a solution but also efficient numerical algorithms for approximating the solution. The theory and methodology of operator equations have now played a very important role in computational mathematics, applied sciences and engineering.

We develop in this paper a theoretical framework for the analysis of convergence for the Petrov-Galerkin method and superconvergence for the iterated Petrov-Galerkin method for Fredholm integral equations of the second kind. As important approaches to the analysis, we introduce notions of the generalized best approximation and the regular pair of trial space sequence and test space sequence. In Hilbert spaces, we characterize the regular pair in terms of the angle of two space sequences or the generalized best approximation projections. Several specic constructions of the Petrov-Galerkin elements for equations of both one dimension and two dimensions are presented and the convergence of the Petrov-Galerkin method and the iterated Petrov-Galerkin method using these elements is proved.

We introduce the concept of a re nable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a re nable set parallels that of a re nable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a re nable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of re nable sets which can be used for generating interpolatory wavelets are included.