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In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval: , = ¯ ().
A Fourier series (/ ˈ f ʊr i eɪ,-i ər / [1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are ...
The Fourier series is a method of expressing a periodic function in terms of sinusoidal basis functions. Taking C[−π,π] to be the space of all real-valued functions continuous on the interval [−π,π] and taking the inner product to be , = ()
Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions.
This sum is called a Chebyshev series or a Chebyshev expansion. Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart. [16] These attributes include: The Chebyshev polynomials form a complete orthogonal system.
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).
A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function f. Hilbert space methods provide one possible answer to this question. [46] The functions e n (θ) = e 2πinθ form an orthogonal basis of the Hilbert space L 2 ([0, 1]).
which can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (see e.g. discrete Fourier transform). Another important property, the inversion formula, involves the Hurwitz zeta function or the Bernoulli polynomials and is found under relationship to other functions below.