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While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically it can be interpreted as a rotation by 90° in the time–frequency domain (considering time as the x-axis and frequency as the y-axis), and the Fourier transform ...
Analogously to the classical Fourier transform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis. The Graph Fourier transform is important in spectral graph theory. It is widely applied in the recent study of graph structured learning algorithms, such as the widely employed convolutional networks.
An Intuitive Explanation of Fourier Theory by Steven Lehar. Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 6 is on the 1- and 2-D Fourier Transform. Lectures 7–15 make use of it., by Alan Peters; Moriarty, Philip; Bowley, Roger (2009). "Σ Summation (and Fourier Analysis)". Sixty ...
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain).
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.
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
Mathematically, the duality between position and momentum is an example of Pontryagin duality. In particular, if a function is given in position space, f(r), then its Fourier transform obtains the function in momentum space, φ(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function.
The lower right corner depicts samples of the DTFT that are computed by a discrete Fourier transform (DFT). The utility of the DTFT is rooted in the Poisson summation formula, which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform: [b]