# Frame approximation with bounded coefficients

@article{Adcock2021FrameAW,
title={Frame approximation with bounded coefficients},
year={2021},
volume={47},
pages={1-25}
}
• Published 3 January 2020
• Computer Science, Mathematics
Due to their flexibility, frames of Hilbert spaces are attractive alternatives to bases in approximation schemes for problems where identifying a basis is not straightforward or even feasible. Computing a best approximation using frames, however, can be challenging since it requires solving an ill-conditioned linear system. One consequence of this ill-conditioning is that the coefficients of such a frame approximation can grow large. In this paper, we resolve this issue by introducing two… Expand

#### References

SHOWING 1-10 OF 17 REFERENCES
Frames and numerical approximation
• Mathematics, Computer Science
• SIAM Rev.
• 2019
The analysis suggests that frames are a natural generalization of bases in which to develop numerical approximation, and even in the presence of severe ill-conditioning, frames impose sufficient mathematical structure so as to give rise to good accuracy in finite precision calculations. Expand
Frames and Numerical Approximation II: Generalized Sampling
• Mathematics
• 2018
In a previous paper [Adcock & Huybrechs, 2019] we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormousExpand
APPROXIMATING SMOOTH, MULTIVARIATE FUNCTIONS ON IRREGULAR DOMAINS
• Mathematics
• Forum of Mathematics, Sigma
• 2020
In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can beExpand
Function Approximation on Arbitrary Domains Using Fourier Extension Frames
• Mathematics, Computer Science
• SIAM J. Numer. Anal.
• 2018
It is shown that for most 2D domains in the fully discrete case the plunge region scales like ${\mathcal O}(N \log N)$, proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. Expand
A Fast Algorithm for Fourier Continuation
• M. Lyon
• Mathematics, Computer Science
• SIAM J. Sci. Comput.
• 2011
A novel decoupling of the least-squares problem is demonstrated which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system. Expand
An introduction to frames and Riesz bases
Frames in Finite-dimensional Inner Product Spaces.- Infinite-dimensional Vector Spaces and Sequences.- Bases.- Bases and their Limitations.- Frames in Hilbert Spaces.- Tight Frames and Dual FrameExpand
Spectral domain embedding for elliptic PDEs in complex domains
Spectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially asExpand
Fast Algorithms for the Computation of Fourier Extensions of Arbitrary Length
• Mathematics, Computer Science
• SIAM J. Sci. Comput.
• 2016
Two $\mathcal{O}(N\log^2N)$ algorithms are presented for the computation of these approximations for the case of general $T$, made possible by exploiting the connection between Fourier extensions and Prolate Spheroidal Wave theory. Expand
Fourier embedded domain methods: extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and C∞
• J. Boyd
• Mathematics, Computer Science
• Appl. Math. Comput.
• 2005
A simple way to solve a partial differential equation in a non-rectangular domain @W is to embed the domain in a rectangle B and solve the problem (more easily) in the rectangle. To apply a FourierExpand
A Sharp-Interface Active Penalty Method for the Incompressible Navier–Stokes Equations
• Mathematics, Physics
• J. Sci. Comput.
• 2015
It is demonstrated that one may achieve high order accuracy by introducing an active penalty term, and one key difference from other works is that it uses a sharp, unregularized mask function. Expand