This paper is concerned with the construction and the analysis of Gauss quadrature formulas for computing integrals of (smooth) functions against refinable functions and wavelets. The main goal of this paper is to develop rigorous error estimates for these formulas. For the univariate setting, we derive asymptotic error bounds for a huge class of weight functions including spline functions. We also discuss multivariate quadrature rules and present error estimates for specific nonseparable refinable functions, i.e., for some special box splines.

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.

It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C1 cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C1 cubic spline spaces over a so-called even stratified triangulation.

We investigate the global smoothness preservation by Bernstein operators measured by second-order modulus of smoothness and give a partial answer for a conjecture raised by H. H. Gonska in 1998.

Dynamical optimization is considered as a general conception based on parametrized approaches and supposed to solve procedural tasks in operational research. Following this guideline, continuous parametric descriptions and estimation by inference are extended to a functional self-consistency approach, which we develop in the present work, inspired by the Maximum Likelihood Method in the context of construction heuristics. We focus on the question, once a formal ansatz is made, whether it yields a reasonable solution to a given self-consistency problem. We show how to construct a self-consistency measure analog to the likelihood function and proceed with “functional samples,” which finally have to yield a best-estimate parameter set and the approximate solution's deviation from the true one, which progressively may produce an adequate solution to a given optimization problem.