Frames and Numerical Approximation

Functions of one or more variables are usually approximated with a basis: a complete, linearly independent system of functions that spans a suitable 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 infinite representations with bounded coefficients. While frames are well known 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. For instance, we exhibit a frame which yields simple, high-order approximations of smooth, multivariate functions in arbitrary geometries. A key concern when using frames is that computing a best approximation requires solving an ill-conditioned linear system. Nonetheless, we construct a frame approximation via regularization with bounded condition number (with respect to perturbations in the data), which approximates any function up to an error of order root epsilon, or even of order epsilon with suitable modifications. Here, epsilon is a threshold value that can be chosen by the user. Crucially, rate of decay of the error down to this level is determined by the existence of approximate representations of f in the frame possessing small-norm coefficients. We demonstrate the existence of such representations in all of our examples. Overall, our analysis suggests that frames are a natural generalization of bases in which to develop numerical approximations. In particular, even in the presence of severely ill-conditioned linear systems, the frame condition imposes sufficient mathematical structure in order to give rise to accurate, well-conditioned approximations.