Stefan Kunis

Using NFFT 3---A Software Library for Various Nonequispaced Fast Fourier Transforms

NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, for example, nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This article provides a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.

Random Sampling of Sparse Trigonometric Polynomials, II. Orthogonal Matching Pursuit versus Basis Pursuit

Abstract We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.

Fast spherical Fourier algorithms

Spherical Fourier series play an important role in many applications. A numerically stable fast transform analogous to the fast Fourier transform is of great interest. For a standard grid of O(N2) points on the sphere, a direct calculation has computational complexity of O(N4), but a simple separation of variables reduces the complexity to O(N3). Here we improve well-known fast algorithms for the discrete spherical Fourier transform with a computational complexity of O(N2 log2 N). Furthermore we present, for the first time, a fast algorithm for scattered data on the sphere. For arbitrary O(N2) points on the sphere, a direct calculation has a computational Complexity of O(N4), but we present an approximate algorithm with a computational complexity of O(N2 log2 N).

Stability Results for Scattered Data Interpolation by Trigonometric Polynomials

A fast and reliable algorithm for the optimal interpolation of scattered data on the torus

Recompression Techniques for Adaptive Cross Approximation

\mathbb{T}^d

A Note on the Iterative MRI Reconstruction from Nonuniform k-Space Data

by multivariate trigonometric polynomials is presented. The algorithm is based on a variant of the conjugate gradient method in combination with the fast Fourier transforms for nonequispaced nodes. The main result is that under mild assumptions the total complexity for solving the interpolation problem at

On the stability of the hyperbolic cross discrete Fourier transform

M

Interpolation lattices for hyperbolic cross trigonometric polynomials

arbitrary nodes is of order

Nonequispaced Hyperbolic Cross Fast Fourier Transform

{\cal O}(M\log M)

Fast Summation of Radial Functions on the Sphere

. This result is obtained by the use of localized trigonometric kernels where the localization is chosen in accordance with the spatial dimension