Convolution theoremIn mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise 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). Other versions of the convolution theorem are applicable to various Fourier-related transforms. Consider two functions and with Fourier transforms and : where denotes the Fourier transform operator.
Terence TaoTerence Chi-Shen Tao (; born 17 July 1975) is an Australian mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. Tao was born to ethnic Chinese immigrant parents and raised in Adelaide.
Noncommutative harmonic analysisIn mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact.
Poisson summation formulaIn mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function.
Locally compact abelian groupIn several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the discrete topology), or the real numbers or the circle (both with their usual topology) are locally compact abelian groups. A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff; the topological group is called abelian if the underlying group is abelian.
