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Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized in the frequency spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component.
Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition. In general, ξ {\displaystyle \xi } must always be taken to be a linear form on the space of its domain, which is to say that the second real line is the dual space of the first real line.
Periodic functions can be identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by or . The Fourier transform is also part of Fourier analysis , but is defined for functions on R n {\displaystyle \mathbb {R} ^{n}} .
When a function () is a function of time and represents a physical signal, the transform has a standard interpretation as the frequency spectrum of the signal. The magnitude of the resulting complex-valued function S ( f ) {\displaystyle S(f)} at frequency f {\displaystyle f} represents the amplitude of a frequency component whose initial phase ...
If f is an odd function with period , then the Fourier Half Range sine series of f is defined to be = = which is just a form of complete Fourier series with the only difference that and are zero, and the series is defined for half of the interval.
Left: A continuous function (top) and its Fourier transform (bottom). Center-left: Periodic summation of the original function (top). Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series. Center-right: Original function is discretized (multiplied by a Dirac comb) (top).
All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation about the sawtooth is called the Gibbs phenomenon. There are many known sufficient conditions for the Fourier series of a function to converge at a given point x, for example if the function is differentiable at x.
Jean-Baptiste Joseph Fourier (/ ˈ f ʊr i eɪ,-i ər /; [1] French: [ʒɑ̃ batist ʒozɛf fuʁje]; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre, Burgundy and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and ...