Frames and numerical approximation

Functions of one or more variables are usually approximated with a basis; a complete, linearly independent set of functions that spans an appropriate function space. The topic of this paper is the numerical approximation of functions using the more general notion of frames; that is, complete systems that are generally redundant but provide stable infinite representations. While frames are well-known tools in image and signal processing, coding theory and other areas of applied mathematics, their use in numerical analysis is far less widespread. Yet, as we show via a series of examples, frames are more flexible than bases, and can be constructed easily in a range of problems where finding orthonormal bases with desirable properties (rapid convergence, high resolution power, etc) is difficult or impossible. The examples highlight three general and useful ways in which frames naturally arise: namely, restrictions of orthonormal bases to subdomains, augmentation of orthonormal bases by a finite number of ‘feature’ functions and concatenation of two or more orthonormal bases. By means of application, we show an example of the first construction which yields simple, high-order approximations of smooth, multivariate functions in complicated geometries. A major concern when using frames is that computing a best approximation typically requires solving an ill-conditioned linear system. Nonetheless, we show that an accurate frame approximation of a function f can be computed numerically up to an error of order √ with a simple algorithm, or even of order with modifications to the algorithm. Here, is a threshold value that can be chosen by the user. Crucially, the order of convergence down to this limit is determined by the existence of accurate, approximate representations of f in the frame that have small-norm coefficients. We demonstrate the existence of such representations in all our examples. Overall, our analysis suggests that frames are a natural generalization of bases in which to develop numerical approximation. In particular, even in the presence of severe illconditioning, the frame condition imposes sufficient mathematical structure in order to give rise to good accuracy in finite precision calculations.