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The Fourier inversion theorem holds for all continuous functions that are absolutely integrable (i.e. ()) with absolutely integrable Fourier transform. This includes all Schwartz functions, so is a strictly stronger form of the theorem than the previous one mentioned.
The characteristic function is closely related to the Fourier transform: the characteristic function of a probability density function p(x) is the complex conjugate of the continuous Fourier transform of p(x) (according to the usual convention; see continuous Fourier transform – other conventions).
Continuous Fourier transform; Fourier inversion theorem; Plancherel's theorem; Convolution; Convolution theorem; Positive-definite function; Poisson summation formula; Paley-Wiener theorem; Sobolev space; Time–frequency representation; Quantum Fourier transform
Download as PDF; Printable version ... by Cauchy's integral theorem. Therefore, the Fourier inversion formula can use integration along different lines, parallel to ...
Download as PDF; Printable version ... Pages in category "Theorems in Fourier analysis" The following 18 pages are in this category, out of 18 total. ... Final value ...
List of Fourier-related transforms; Fourier transform on finite groups; Fractional Fourier transform; Continuous Fourier transform; Fourier operator; Fourier inversion theorem; Sine and cosine transforms; Parseval's theorem; Paley–Wiener theorem; Projection-slice theorem; Frequency spectrum
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Fourier inversion theorem, any one of several theorems by which Fourier inversion recovers a function from its Fourier transform; Short-time Fourier transform or short-term Fourier transform (STFT), a Fourier transform during a short term of time, used in the area of signal analysis