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Intra-MRI Head Motion Tracking and Correction: A Quantitative In Vivo Evaluation Framework. [PDF]

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The Fourier transform and the discrete Fourier transform

Inverse Problems, 1989
This paper gives an error bound in computing the Fourier transform for a square summable function by means of the discrete Fourier transform. In detail description, the error bound depends on the number of samples, the interval where the samples are taken, the interval where the Fourier transform is being approximated, the local averaging in the time ...
Auslander, Louis, Grünbaum, F. Alberto
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On Fourier Series on the Torus and Fourier Transforms

Mathematical Notes, 2021
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The Fourier Transform

Scientific American, 1989
To calculate a transform, just listen. The ear automatically performs the calculation, which the intellect can execute only after years of mathematical education. The ear formulates a transform by converting sound-the waves of pressure traveling through time and the atmosphere-into a spectrum, a description of the sound as a series of volumes at ...
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On Fourier-Stieltjes Transforms

Canadian Journal of Mathematics, 1955
Let be the class of bounded non-decreasing functions defined on the real line which are normalized by the conditions ϕ(− ∞) = 0 , ϕ(t + 0) = ϕ(t).Let be the class of Fourier-Stieltjes transforms of elements of i.e. the elements of and are connected by the relationwhere ϕ ∊ and Φ ∊ .It is well known, and easy to verify that this mapping from to ...
Calderón, Alberto P., Devinatz, A.
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Accuracy of the Discrete Fourier Transform and the Fast Fourier Transform

SIAM Journal on Scientific Computing, 1996
Accuracy of the discrete Fourier transform (DFT) and the fast Fourier transform (FFT) depends on the accuracy of the twiddle factors entirely. For accurate twiddle factor tables, this paper recommends to compute the sine/cosine functions with high precision arithmetic along the algorithms in terms of faster converging approximations, such as rational ...
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