Daniel Potts

Fast Fourier Transforms for Nonequispaced Data: A Tutorial

In this chapter we consider approximativemethods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particularwe are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTiaalgorithms with respect to roundoff errors and applyNDFTalgorithms for the fast computation of Besseltransforms.

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.

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).

Fast algorithms for discrete polynomial transforms

Consider the Vandermonde-like matrix P:= (P k (cos jπ/N)) j,k=0 N , where the polynomials P k satisfy a three-term recurrence relation. If P k are the Chebyshev polynomials T k , then P coincides with C N+1 := (cos jkπ/N) j,k=0 N . This paper presents a new fast algorithm for the computation of the matrix-vector product Pa in O(N log 2 N) arithmetical operations. The algorithm divides into a fast transform which replaces Pa with C N+1 ā and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes Pa with almost the same precision as the Clenshaw algorithm, but is much faster for N ≥ 128.

Parameter estimation for exponential sums by approximate Prony method

The recovery of signal parameters from noisy sampled data is a fundamental problem in digital signal processing. In this paper, we consider the following spectral analysis problem: Let f be a real-valued sum of complex exponentials. Determine all parameters of f, i.e., all different frequencies, all coefficients, and the number of exponentials from finitely many equispaced sampled data of f. This is a nonlinear inverse problem. In this paper, we present new results on an approximate Prony method (APM) which is based on [1]. In contrast to [1], we apply matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail. The number of sampled data acts as regularization parameter. The first part of APM estimates the frequencies and the second part solves an overdetermined linear Vandermonde-type system in a stable way. We compare the first part of APM also with the known ESPRIT method. The second part is related to the nonequispaced fast Fourier transform (NFFT). Numerical experiments show the performance of our method.

Stability Results for Scattered Data Interpolation by Trigonometric Polynomials

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

Fast convolution with radial kernels at nonequispaced knots

\mathbb{T}^d

OPTIMAL TRIGONOMETRIC PRECONDITIONERS FOR NONSYMMETRIC TOEPLITZ SYSTEMS

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

Nonlinear Approximation by Sums of Exponentials and Translates

M

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

arbitrary nodes is of order