Fourier Analysis  

Fourier Series / Components of a Function
$ 1-dimensional: For f : [−L, L] → \(\mathbb C\),

f(x) = (2L)−1/2k = −∞+∞ fk exp{iπ kx/L},   with   fk:= (2L)−1/2 −LL dx f(x) exp{−iπ kx/L} .

$ n-dimensional: For f : Tn → \(\mathbb C\), i.e., φi ∈ [0, 2π], obtained replacing πx/L by φ in the 1D case,

f(φ) = ∑k ∈ \(\mathbb Z\)n  fk exp{i k · φ},   with   fk:= (2π)n Tn dnφ f(φ) exp{−i k · φ} .

* Wiener's theorem: If f(t) = ∑n=−∞ cn exp{2πi tn} is a non-zero absolutely convergent (Fourier) series on the unit interval [0,1], then 1/f(t) can be represented as an absolutely convergent Fourier series on this interval.
> Online resources: see MathWorld page; Wikipedia page.

Fourier Transform of a Function > s.a. Convolution; Gel'fand Transform; Wavelets [alternative].
* Idea: A map f \(\mapsto\) \(\cal F\)(f) = \(\tilde f\) that gives the frequency distribution in f ; If f is in L2(\(\mathbb R\)n), then so is \(\tilde f\).
$ Def: The Fourier transform of f and its inverse Fourier transform are, respectively,

\(\tilde f\)(k):= (2π)n/2 \(\,_{{\mathbb R}^n}^~\) dnx f(x) exp{−i k · x},   and   f(x) = (2π)n/2 \(\,_{{\mathbb R}^n}^~\) dnk \(\tilde f\)(k) exp{i k · x}.

* Examples: For the Dirac delta function,

δ(n)(xx0) = (2π)n\(\int_{-\infty}^{+\infty}\) dk exp{i k · (xx0)} ;

For other common functions,

\(\,_{{\mathbb R}^n}^~\) d3x exp{−i k · x} r−1 = 4π/k2 ,   \(\,_{{\mathbb R}^n}^~\) d3k exp{i k · x} k−2 = 2π2/r ;

For a Gaussian f(x) = N exp{−x2/2σ2}, the Fourier transform \(\tilde f\)(k) = N' exp{−k2σ2/2} [the proof for σ = 1 is that the function f(x) = N exp{−x2/2} is the general solution of the ordinary differential equation x f(x) + f '(x) = 0, and the Fourier transform of that differential equation is the same equation – see Rudin].
* Properties: The Fourier transform of a product of two functions is the convolution of their Fourier transforms.
* Numerical calculation: For routines, see IMSL library or CERN library; Defined by

yj = N−1/2k xk exp{2πi jk/N},   k = 0, ..., N−1 ,   j = 0, ..., N .

* Analytic extensions: If F(z) is the analytic extension of f(x), then f(x + i y) is in L2(\(\mathbb R\),dx) for some y > 0 iff \(\tilde f\)(k) vanishes for k < 0, and in that case \(\tilde f\)(k,y) = exp{−ky} \(\tilde f\)(k).
* Generalizations: The fractional Fourier transform requires only a symplectic structure on phase space, not a linear structure.
@ General references: Oberhettinger 73, 73 [tables]; Brigham 74; Bracewell 99 [and applications]; James 11 [guide and applications, II].
@ Fractional Fourier transform: Ozaktas et al AIEP(99), 01; Chmielowiec & Kijowski JGP(12)-a1002 [and geometric quantization]; Coftas & Dragoman JPA(13)-a1301 [discrete]; Chen & Fan a1307.
@ Other generalizations: Yang AITS-a1106 [Yang-Fourier transform in fractal space]; Plastino & Rocca PhyA(12)-a1112 [complex q-Fourier transform]; De Bie MMAS(12)-a1209 [generalizations related to the Lie algebra sl\(_2\) and the Lie superalgebra osp(1|2)]; Oriti & Rosati PRD(19)-a1812 [non-commutative]; Horwitz a1907 [on a manifold].
@ Special topics: Kempf JMP(00)gq/99 [finite bandwidth and rapid variation]; Wurm et al JMP(03) [Lorentz-invariant functions]; McCallum & Horikis JPA(06) [self-Fourier functions]; Carley a1310/JPA [Bessel function multiplied by a Gaussian]; > s.a. representations in quantum theory.
> Online resources: see MathWorld page; Wikipedia page.

Related Concepts and Results > s.a. Superoscillations.
$ Parseval's relation / theorem: If F(k) and G(k) are the Fourier transforms of f(x) and g(x), respectively, then

dx f(x) g*(x) = dk F(k) G*(k) .

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