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Nonuniform fast Fourier transform
 Geophysics
, 1999
"... The nonuniform discrete Fourier transform (NDFT) can be computed with a fast algorithm, referred to as the nonuniform fast Fourier transform (NFFT). In L dimensions, the NFFT requires O(N(ln #) L + ( Q L #=1 M # ) P L #=1 log M # ) operations, where M # is the number of Fourier components ..."
Abstract

Cited by 44 (1 self)
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The nonuniform discrete Fourier transform (NDFT) can be computed with a fast algorithm, referred to as the nonuniform fast Fourier transform (NFFT). In L dimensions, the NFFT requires O(N(ln #) L + ( Q L #=1 M # ) P L #=1 log M # ) operations, where M # is the number of Fourier components along dimension #, N is the number of irregularly spaced samples, and # is the required accuracy. This is a dramatic improvement over the O(N Q L #=1 M # ) operations required for the direct evaluation (NDFT). The performance of the NFFT depends on the lowpass filter used in the algorithm. A truncated Gauss pulse, proposed in the literature, is optimized. A newly proposed filter, a Gauss pulse tapered with a Hanning window, performs better than the truncated Gauss pulse and the Bspline, also proposed in the literature. For small filter length, a numerically optimized filter shows the best results. Numerical experiments for 1D and 2D implementations confirm the theoretically predicted ...
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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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,