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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).
The original proof by Titchmarsh uses complex-variable techniques, and is based on the Phragmén–Lindelöf principle, Jensen's inequality, Carleman's theorem, and Valiron's theorem. The theorem has since been proven several more times, typically using either real-variable [3] [4] [5] or complex-variable [6] [7] [8] methods.
Download as PDF; Printable version; In other projects Wikidata item; Appearance. move to sidebar hide. Help ... Titchmarsh convolution theorem; W. Walsh–Lebesgue ...
A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional unitary representations form an orthonormal basis in L 2 by the Peter–Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects of harmonic analysis that depend on the Fourier transform.
In mathematics, particularly in the area of Fourier analysis, the Titchmarsh theorem may refer to: The Titchmarsh convolution theorem; The theorem relating real and imaginary parts of the boundary values of a H p function in the upper half-plane with the Hilbert transform of an L p function. See Hilbert transform#Titchmarsh's theorem
Download as PDF; Printable version; ... Then per the convolution theorem, the starred transform is equivalent to the complex convolution of [()] = ...
Download as PDF; Printable version; ... (or Vandermonde's convolution) ... There is a q-analog to this theorem called the q-Vandermonde identity.
Download QR code; Print/export Download as PDF; Printable version; In other projects ... Convolution theorem; Analysis of unevenly spaced data