Multivariate Approximation

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Approximation theory studies the ability to approximate general, possibly complicated, functions by using simpler and more easily calculated functions. Many problems in approximation theory of functions of one variable have been solved. However, the methods do not always easily generalize to the approximation of functions of several variables. The main goal of my proposed research is to improve existing and create new methods for the approximation of functions of several variables. I will focus on questions related to measures of smoothness (how can we easily measure or "predict" the accuracy of approximation), inequalities for polynomials (useful tools for proving various theoretical results) and shape preserving approximation (if the target function or object has a certain geometric shape such as convexity, how to construct an approximation which also possesses such a shape). In the proposed research, the concept of convexity appears in different contexts and many arguments have very geometric flavour. There will be numerous opportunities for training of highly qualified personnel.*******The research has primarily a theoretical flavour, and it is anticipated that the results will be applicable in approximation theory, harmonic analysis, functional analysis, convex geometry and other areas of mathematics. Some outcomes will also be useful for numerical analysis and computer-aided geometric design. **