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Random sampling of multivariate trigonometric polynomials
 SIAM J. Math. Anal
, 2004
"... We investigate when a trigonometric polynomial p of degree M in d variables is uniquely determined by its sampled values p(xj) on a random set of points xj in the unit cube (the “sampling problem for trigonometric polynomials”) and estimate the probability distribution of the condition number for th ..."
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Cited by 31 (3 self)
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We investigate when a trigonometric polynomial p of degree M in d variables is uniquely determined by its sampled values p(xj) on a random set of points xj in the unit cube (the “sampling problem for trigonometric polynomials”) and estimate the probability distribution of the condition number for the associated Vandermondetype and Toeplitzlike matrices. The results provide a solid theoretical foundation for some efficient numerical algorithms that are already in use.
Using NFFT 3  a software library for various nonequispaced fast Fourier transforms
, 2008
"... NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and ..."
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Cited by 12 (8 self)
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NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide 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.
Proceedings of Symposia in Applied Mathematics FanBeam Tomography and Sampling Theory
"... Abstract. Computed tomography entails the reconstruction of a function from measurements of its line integrals. In this article we explore the question: How many and which line integrals should be measured in order to achieve a desired resolution in the reconstructed image? Answering this question m ..."
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Abstract. Computed tomography entails the reconstruction of a function from measurements of its line integrals. In this article we explore the question: How many and which line integrals should be measured in order to achieve a desired resolution in the reconstructed image? Answering this question may help to reduce the amount of measurements and thereby the radiation dose, or to obtain a better image from the data one already has. Our exploration leads us to a mathematically and practically fruitful interaction of Shannon sampling theory and tomography. For example, sampling theory helps to identify efficient data acquisition schemes, provides a qualitative understanding of certain artifacts in tomographic images, and facilitates the error analysis of some reconstruction algorithms. On the other hand, applications in tomography have stimulated new research in sampling theory, e.g., on nonuniform sampling theorems and estimates for the aliasing error. The focus of this article will be the application of sampling theory to the socalled fanbeam geometry. Its dual aim is an exposition of the main principles involved as well as the development of some new insights. 1.
Contents
"... 2 Notation, the NDFT, and the NFFT 5 2.1 NDFT nonequispaced discrete Fourier transform............... 5 2.2 NFFT nonequispaced fast Fourier transform.................. 6 ..."
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2 Notation, the NDFT, and the NFFT 5 2.1 NDFT nonequispaced discrete Fourier transform............... 5 2.2 NFFT nonequispaced fast Fourier transform.................. 6