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Fast Fourier transforms for nonequispaced data: A tutorial
, 2000
"... In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity o ..."
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Cited by 111 (33 self)
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In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTalgorithms with respect to roundoff errors and apply NDFTalgorithms for the fast computation of Bessel transforms.
Stability Results for Scattered Data Interpolation by Trigonometric Polynomials
 SIAM J. Sci. Comput
, 2007
"... A fast and reliable algorithm for the optimal interpolation of scattered data on the torus Td 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 ..."
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Cited by 31 (17 self)
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A fast and reliable algorithm for the optimal interpolation of scattered data on the torus Td 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 M arbitrary nodes is of order O(M log M). This result is obtained by the use of localised trigonometric kernels where the localisation is chosen in accordance to the spatial dimension d. Numerical examples show the efficiency of the new algorithm.
A MultiLevel Algorithm for the Solution of Moment Problems
, 1997
"... We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multilevel algorithms, based on the conjugate gradient method and the LandweberRichardson method are proposed, that determines the "optimal ..."
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Cited by 3 (1 self)
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We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multilevel algorithms, based on the conjugate gradient method and the LandweberRichardson method are proposed, that determines the "optimal" reconstruction level a posteriori from quantities that arise during the numerical calculations. As an important example we discuss the reconstruction of bandlimited signals from irregularly spaced noisy samples, when the actual bandwidth of the signal is not available. Numerical examples show the usefulness of the proposed algorithms. Keywords and Phrases: Moment problems, multilevel algorithms, Landweber Richardson method, conjugate gradient method, sampling theory. AMS Subject Classification: 44A60, 65J10, 65J20, 62D05, 42A15. 1 Introduction We study numerical methods for the solution of a family of general linear moment problems hx; g N j i = j j 2 I (1.1) where fg N j g j2I is...
Reconstruction of periodic bandlimited signals from nonuniform samples
, 2004
"... I would first like to thank my advisor, Dr. Yonina Eldar, for her support, patience, and shoving me in the right direction in critical moments. Thank you for the knowledge I acquired from you. I feel extremely fortunate to have an advisor like you. Thanks to Prof. Arie Feuer of the Technion–Israel I ..."
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Cited by 3 (2 self)
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I would first like to thank my advisor, Dr. Yonina Eldar, for her support, patience, and shoving me in the right direction in critical moments. Thank you for the knowledge I acquired from you. I feel extremely fortunate to have an advisor like you. Thanks to Prof. Arie Feuer of the Technion–Israel Institute of Technology for first introducing me the field of nonuniform sampling. The initial step for this research was carried out from these fruitful discussions. Thanks to Prof. Amir Averbuch of the Tel Aviv University for raising the issue of stability of the algorithms proposed in this work. This topic stimulated the significant phase of my research. I want to also thank Prof. Michael Unser and Dr. Thierry Blu from EPFL, Swiss, and Dr. Thomas Strohmer from University of California, Davis for valuable suggestions related to this work. Thanks to every member of the research group under the supervision of Dr. Yonina Eldar. Ami, Liron, Zvika, Tsvika, Nagesh, Moshe, and Noam, I thank you all for your friendship, and for all the help through the endless discussions we had.
A LEVINSONGALERKIN ALGORITHM FOR REGULARIZED TRIGONOMETRIC APPROXIMATION ∗
, 1999
"... Abstract. Trigonometric polynomials are widely used for the approximation of a smooth function f from a set of nonuniformly spaced samples {f(xj)} N−1 j=0. If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfit ..."
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Cited by 1 (0 self)
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Abstract. Trigonometric polynomials are widely used for the approximation of a smooth function f from a set of nonuniformly spaced samples {f(xj)} N−1 j=0. If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Using the polynomial degree as regularization parameter we derive a multilevel algorithm that iteratively adapts to the least squares solution of optimal smoothness. The proposed algorithm computes the solution in at most O(NM +M 2) operations (M being the polynomial degree of the approximation) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the Left Ventricle.