In this talk, we describe a method of approximating mixed Hölder functions using random samples via a construction that involves dyadic rectangles and random matrix theory. An example of a function class that satisfies a mixed Hölder condition is the space of real-valued functions f on the unit square whose mixed partial derivative ∂^2 f/(∂ x ∂ y) is bounded. This bounded mixed derivative condition is not rotationally invariant, which limits the analytical tools that are available. One approach, due to Smolyak, is to represent these functions by linear combinations of indicator functions of dyadic rectangles. We present a randomized version of Smolyak's method in two dimensions. We also discuss ongoing work in higher dimensions, and applications to matrix organization. 

Nicholas Marshall is an applied mathematics PhD student at Yale University advised by Raphy Coifman and Stefan Steinerberger. He is interested in problems that involve an interplay between analysis and geometry, and applications of all types.