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The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the ...
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Fourier and related algebras occur naturally in the harmonic analysis of locally compact groups.They play an important role in the duality theories of these groups. The Fourier–Stieltjes algebra and the Fourier–Stieltjes transform on the Fourier algebra of a locally compact group were introduced by Pierre Eymard in 1964.
Analogously to the classical Fourier transform, graph Fourier transform provides a way to represent a signal in two different domains: the vertex domain and the graph spectral domain. Note that the definition of the graph Fourier transform and its inverse depend on the choice of Laplacian eigenvectors, which are not necessarily unique. [3]
A function in the Schwartz space is sometimes called a Schwartz function. A two-dimensional Gaussian function is an example of a rapidly decreasing function. Schwartz space is named after French mathematician Laurent Schwartz .
Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze spatial frequencies, and indeed for nearly any function domain. This justifies their use in such diverse branches as image processing, heat conduction, and automatic control.
Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized combinatorially by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces.
Left: A continuous function (top) and its Fourier transform (bottom). Center-left: Periodic summation of the original function (top). Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series. Center-right: Original function is discretized (multiplied by a Dirac comb) (top).