Driven by its numerous applications in computational science, the approximation of smooth, high-dimensional functions via sparse polynomial expansions has received significant attention in the last five to ten years. In the first part of this talk, I will give a brief survey of recent progress in this area. In particular, I will demonstrate how the proper use of compressed sensing tools leads to new techniques for high-dimensional approximation which can mitigate the curse of dimensionality to a substantial extent. The rest of the talk is devoted to approximating functions defined on irregular domains. The vast majority of works on high-dimensional approximation assume the function in question is defined over a tensor-product domain. Yet this assumption is often unrealistic. I will introduce a method, known as polynomial frame approximation, suitable for broad classes of irregular domains and present theoretical guarantees for its approximation error, stability, and sample complexity. These results show the suitability of this approach for high-dimensional approximation through the independence (or weak dependence) of the various guarantees on the ambient dimension d. Time permitting, I will also discuss several extensions.
