Book Review: Fourier transforms

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Sparse Generalized Fourier Transforms ∗

Block-diagonalization of sparse equivariant discretization matrices is studied. Such matrices typically arise when partial differential equations that evolve in symmetric geometries are discretized via the finite element method or via finite differences. By considering sparse equivariant matrices as equivariant graphs, we identify a condition for when block-diagonalization via a sparse variant ...

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Quadratic Fourier Transforms

In this paper we shall examine the quadratic Fourier transform which is introduced by the generalized quadratic function for one order parameter in the ordinary Fourier transform. This will be done by analyzing corresponding six subcases of the quadratic Fourier transform within a reproducing kernel Hilbert spaces framework. Center for R&D in Mathematics and Applications, Department of Mathemat...

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Fast Fourier Transforms

29 O(b log(b)) operations (using standard multiplication). As there are O(b= log(b)) primes in total, the running time of this stage of the algorithm is O(b 2 L), even using the \grammar school" method of integer multiplication. At this stage of the algorithm we have obtained a vector of length L whose entries are integral linear combinations of powers of with coeecients bounded by M in absolut...

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Dimensionless Fast Fourier Transforms

This paper shows that there are fast Fourier transform (FFT) algorithms that work, for a fixed number of points, independent of the dimension. Changing the dimension is achieved by relabeling the input and the output and changing the “twiddle factors.” An important consequence of this result, is that a program designed to compute the 1-dimensional Fourier transform can be easily modified to com...

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Fuzzy and Fourier Transforms

The fuzzy transform (F -transform for short) is a universal tool for a fuzzy modeling with convincing applications to image processing. The aim of this contribution is to explain the effect of the F transform in image processing. With this purpose, we investigate properties of the Fourier transform over the F -transform components. We prove that the direct F -transform is a low-pass filter. Thi...

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ژورنال

عنوان ژورنال: Bulletin of the American Mathematical Society

سال: 1961

ISSN: 0002-9904

DOI: 10.1090/s0002-9904-1961-10616-2