Fourier Analysis| Fourier Analysis |
Fourier Series / Components of a Function
$ 1-dimensional: For f :
[−L, L] → \(\mathbb C\),
f(x) = (2L)−1/2 ∑k = −∞+∞ 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)(x − x0) = (2π)−n\(\int_{-\infty}^{+\infty}\) dk exp{i k · (x − x0)} ;
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−1/2 ∑k 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) .
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 9 jul 2019