Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Fast Fourier transforms for nonequispaced data
SIAM Journal on Scientific Computing
Rational trigonometric approximations using Fourier series partial sums
Journal of Scientific Computing
SIAM Review
The Regular Fourier Matrices and Nonuniform Fast Fourier Transforms
SIAM Journal on Scientific Computing
A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds
Journal of Computational Physics
Fast monte-carlo algorithms for finding low-rank approximations
Journal of the ACM (JACM)
Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix
SIAM Journal on Computing
SIAM Journal on Computing
Journal of Computational Physics
On the Fourier Extension of Nonperiodic Functions
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Nonuniform fast Fourier transforms using min-max interpolation
IEEE Transactions on Signal Processing
Approximation error in regularized SVD-based Fourier continuations
Applied Numerical Mathematics
On the resolution power of Fourier extensions for oscillatory functions
Journal of Computational and Applied Mathematics
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A new algorithm is presented which provides a fast method for the computation of recently developed Fourier continuations (a particular type of Fourier extension method) that yield superalgebraically convergent Fourier series approximations of nonperiodic functions. Previously, the coefficients of an approximating Fourier series have been obtained by means of a regularized singular value decomposition (SVD)-based least-squares solution to an overdetermined linear system of equations. These SVD methods are effective when the size of the system does not become too large, but they quickly become unwieldy as the number of unknowns in the system grows. We demonstrate a novel decoupling of the least-squares problem 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. Utilizing randomized algorithms, the low-rank system is reduced to a significantly smaller system of equations. This new system is then efficiently solved with drastically reduced computational cost and memory requirements while still benefiting from the advantages of using a regularized SVD. The computational cost of the new algorithm in on the order of the cost of a single FFT multiplied by a slowly increasing factor that grows only logarithmically with the size of the system.