Approximating high-dimensional functions on irregular domains

Multivariate polynomials are excellent means of approximating high-dimensional functions on tensor-product domains.  But what about approximations on irregular domains, as is quite common in applications?  In our new paper, Daan Huybrechs and I tackle this question via using tools from frame theory and approximation theory:

Approximating smooth, multivariate functions on irregular domains

We establish a series of results on approximation rates and sample complexity, deriving bounds that scale well with dimension d in a variety of cases.