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Joseph Fourier introduced sine and cosine transforms (which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally defined as an improper Riemann integral, making it an integral ...
The Fourier transform is also part ... is the phase shift of the ... equations such as the heat equation, one notable application of Fourier series on the ...
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration ...
Hence, at a particular frequency, the sine transform and the cosine transform together essentially only represent one sine wave that could have any phase shift. An advantage of the modern Fourier transform is that while the sine and cosine transforms together are required to extract the phase information of a frequency, the modern Fourier ...
The short-time Fourier transform (STFT) is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. [1] In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier ...
The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms. [8] In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum.
Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform. In applications of the Fourier transform the Fourier inversion theorem often plays a critical role. In many situations the basic ...
Klibanov, M.V. (1987). "Determination of a function with compact support from the absolute value of its Fourier transform and an inverse scattering problem". Differential Equations. 22: 1232– 1240. Klibanov, M.V. (1987). "Inverse scattering problems and restoration of a function from the modulus of its Fourier transform". Siberian Math. J.