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 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 in a variety of cases.