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12
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.
Fast algorithms for discrete polynomial transforms
 Math. Comput
, 1998
"... Abstract. Consider the Vandermondelike matrix P: = (Pk(cos jπ ..."
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Cited by 27 (7 self)
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Abstract. Consider the Vandermondelike matrix P: = (Pk(cos jπ
Fast and stable algorithms for discrete spherical Fourier transforms
, 1996
"... . In this paper, we propose an algorithm for the stable and efficient computation of Fourier expansions of square integrable functions on the unit sphere S ae R 3 , as well as for the evaluation of these Fourier expansions at special knots. The heart of the algorithm is an efficient realization of ..."
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Cited by 16 (4 self)
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. In this paper, we propose an algorithm for the stable and efficient computation of Fourier expansions of square integrable functions on the unit sphere S ae R 3 , as well as for the evaluation of these Fourier expansions at special knots. The heart of the algorithm is an efficient realization of discrete Legendre function transforms based on a modified and stabilized version of the DriscollHealy algorithm. 1991 Mathematics Subject Classification. Primary 65T99, 33C35, 33C25, 42C10 Key words and phrases. Discrete spherical Fourier transform, spherical harmonics, sampling theorem, discrete Legendre function transform, fast cosine transform, Chebyshev nodes, cascade summation 1 Introduction Fourier analysis on the sphere S ae R 3 has practical relevance in tomography, geophysics, seismology, meteorology and crystallography. It can be used in spectral methods for solving partial differential equations on the sphere (see [4], [16]). In [12], the authors utilize spherical Fourier...
Optimal trigonometric preconditioners for nonsymmetric Toeplitz systems
, 1998
"... . This paper is concerned with the solution of systems of linear equations T N xN = bN , where fT N gN2IN denotes a sequence of nonsingular nonsymmetricToeplitz matrices arising from a generating function of the Wiener class. We present a technique for the fast construction of optimal trigonometric ..."
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Cited by 7 (3 self)
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. This paper is concerned with the solution of systems of linear equations T N xN = bN , where fT N gN2IN denotes a sequence of nonsingular nonsymmetricToeplitz matrices arising from a generating function of the Wiener class. We present a technique for the fast construction of optimal trigonometric preconditioners M N = M N (T 0 N T N ) of the corresponding normal equation which can be extended to Toeplitz least squares problems in a straightforward way. Moreover, we prove that the spectrum of the preconditioned matrix M \Gamma1 N T 0 N T N is clustered at 1 such that the PCGmethod applied to the normal equation converges superlinearly. Numerical tests confirm the theoretical expectations. 1991 Mathematics Subject Classification. 65F10, 65F15, 65T10. Key words and phrases. Toeplitz matrix, Krylov space methods, CGmethod, preconditioners, normal equation, clusters of eigenvalues. 1 Introduction Consider the system of linear equations T N xN = bN ; (1.1) where T N 2 IR N;N...
Preconditioners for illconditioned Toeplitz matrices
, 1999
"... . This paper is concerned with the solution of systems of linear equations ANx = b, where fAN g N2N denotes a sequence of positive definite Hermitian illconditioned Toeplitz matrices arising from a (realvalued) nonnegative generating function f 2 C2ß with zeros. We construct positive definite ..."
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Cited by 5 (3 self)
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. This paper is concerned with the solution of systems of linear equations ANx = b, where fAN g N2N denotes a sequence of positive definite Hermitian illconditioned Toeplitz matrices arising from a (realvalued) nonnegative generating function f 2 C2ß with zeros. We construct positive definite Hermitian preconditioners MN such that the eigenvalues of M \Gamma1 N AN are clustered at 1 and the corresponding PCGmethod requires only O(N log N) arithmetical operations to achieve a prescribed precision. We sketch how our preconditioning technique can be extended to symmetric Toeplitz systems, doubly symmetric block Toeplitz systems with Toeplitz blocks and non Hermitian Toeplitz systems. Numerical tests confirm the theoretical expectations. AMS subject classification: 65F10, 65F15, 65T10. Key words: Illconditioned Toeplitz matrices , CGmethod, clusters of eigenvalues, preconditioners. 1 Introduction. Systems of linear equations ANx = b with positive definite Hermitian T...
Preconditioners for illconditioned Toeplitz systems constructed from positive kernels
, 1999
"... . In this paper, we are interested in the iterative solution of ill{conditioned Toeplitz systems generated by continuous non{negative real{valued functions f with a nite number of zeros. We construct new w{circulant preconditioners without explicit knowledge of the generating function f by approxima ..."
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Cited by 4 (0 self)
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. In this paper, we are interested in the iterative solution of ill{conditioned Toeplitz systems generated by continuous non{negative real{valued functions f with a nite number of zeros. We construct new w{circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f KN with a suitable positive reproducing kernel KN . By the restriction to positive kernels we obtain positive denite preconditioners. Moreover, if f has only zeros of even order 2s, then we can prove that the property R t 2k KN (t) dt CN 2k (k = 0; : : : ; s) of the kernel is necessary and sucient to ensure the convergence of the PCG{method in a number of iteration steps independent of the dimension N of the system. Our theoretical results were conrmed by numerical tests. 1991 Mathematics Subject Classication. 65F10, 65F15, 65T10. Key words and phrases. Ill{conditioned Toeplitz matrices , CG{method, preconditioners, reproducing kernels. 1 Intr...
Numerical stability of fast trigonometric transforms  a worst case study
 J. Concrete Appl. Math
, 2003
"... This paper presents some new results on numerical stability for various fast trigonometric transforms. In a worst case study, we consider the numerical stability of the classical fast Fourier transform (FFT) with respect to different precomputation methods for the involved twiddle factors and show t ..."
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Cited by 4 (2 self)
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This paper presents some new results on numerical stability for various fast trigonometric transforms. In a worst case study, we consider the numerical stability of the classical fast Fourier transform (FFT) with respect to different precomputation methods for the involved twiddle factors and show the strong influence of precomputation errors on the numerical stability of the FFT. The examinations are extended to fast algorithms for the computation of discrete cosine and sine transforms and to efficient computations of discrete Fourier transforms for nonequispaced data. Numerical tests confirm the theoretical estimates of numerical stability.
Fast multidimensional scattered data approximation with Neumann boundary conditions
 Lin. Alg. Appl
, 2004
"... Abstract. An important problem in applications is the approximation of a function f from a finite set of randomly scattered data f(xj). A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials {e 2πikx}. This leads to fast numerical ..."
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Cited by 4 (0 self)
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Abstract. An important problem in applications is the approximation of a function f from a finite set of randomly scattered data f(xj). A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials {e 2πikx}. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials cos(πkx) as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrixvector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a twodimensional geophysical scattered data problem. Key words. Trigonometric approximation, nonuniform sampling, discrete cosine transform,
A fast Fourier algorithm on the rotation group. Preprint A0706, Univ. zu
, 2007
"... In this paper we present an algorithm for the fast Fourier transform on the rotation group SO(3) which is based on the fast Fourier transform for nonequispaced nodes on the threedimensional torus. This algorithm allows to evaluate the SO(3) Fourier transform of Bbandlimited functions at M arbitra ..."
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Cited by 3 (1 self)
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In this paper we present an algorithm for the fast Fourier transform on the rotation group SO(3) which is based on the fast Fourier transform for nonequispaced nodes on the threedimensional torus. This algorithm allows to evaluate the SO(3) Fourier transform of Bbandlimited functions at M arbitrary input nodes in O(M + B 3 log 2 B) flops instead of O(MB 3). Some numerical results will be presented establishing the algorithm’s numerical stability and time requirements.
Trigonometric Preconditioners for Block Toeplitz Systems
, 1997
"... . This paper is concerned with the solution of a system of linear equations TT T T M;N xx x x = bb b b, where TT T T M;N denotes a positive definite doubly symmetric blockToeplitz matrix with Toeplitz blocks arising from a generating function f of the Wiener class. We derive optimal and Strangty ..."
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Cited by 1 (1 self)
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. This paper is concerned with the solution of a system of linear equations TT T T M;N xx x x = bb b b, where TT T T M;N denotes a positive definite doubly symmetric blockToeplitz matrix with Toeplitz blocks arising from a generating function f of the Wiener class. We derive optimal and Strangtype trigonometric preconditioners PP P P M;N of TT T T M;N from the Fej'er and Fourier sum of f , respectively. Using relations between trigonometric transforms and Toeplitz matrices, we prove that for all " ? 0 and sufficiently large M;N , at most O(M)+O(N) eigenvalues of PP P P \Gamma1 M;N TT T T M;N lie outside the interval (1 \Gamma "; 1 + ") such that the preconditioned conjugate gradient method converges in at most O(M)+O(N) steps. x1. Introduction Systems of linear equations TT T TM;N xx x x = bb b b where TT T TM;N denotes a positive definite doubly symmetric blockToeplitz matrix with Toeplitz blocks (BTTB matrices) arise in a variety of applications in mathematics and engineer...