Frames of Hilbert spaces are ubiquitous in image and signal processing, coding theory and sampling theory. However, they are far less widely known in numerical analysis.

In a new paper with Daan Huybrechs, we take a look at frames from a numerical analyst’s perspective:

Frames and numerical approximation

First, we point out that frames can be useful tools in numerical analysis where orthonormal bases may be difficult or impossible to construct. Second, we investigate issues concerning stability and accuracy in frame approximations. Our main result is that frame approximations are stable and accurate, provide the function being approximated has representations in the frame with small-norm coefficients.

One application of this work is meshfree approximation of functions on complex geometries using so-called Fourier extensions. Daan maintains a GitHub page with fast algorithms for computing such approximations.