Fourier Analysis |

**Fourier Series / Components of a Function**

$ __1-dimensional__: For *f* :
[–*L*, *L*] → \(\mathbb C\),

*f*(*x*) = (2*L*)^{–1/2} ∑_{k =
–∞}^{+∞} *f*_{k}
exp{iπ* kx*/*L*}, with *f*_{k}:=
(2*L*)^{–1/2} ∫_{–L}^{L} d*x* *f*(*x*)
exp{–iπ* kx*/*L*} .

$ __n____-dimensional__:
For *f* :
T^{n} → \(\mathbb C\), i.e., *φ*^{i} ∈ [0,
2π],
obtained replacing π*x*/*L* by *φ* in the 1D case,

*f*(*φ*) = ∑_{k ∈
\(\mathbb Z\)n} *f*_{k} exp{i *k* · *φ*}, with *f*_{k}:=
(2π)^{–n} ∫_{ Tn} d^{n}*φ** f*(*φ*)
exp{–i *k* · *φ*} .

* __Wiener's theorem__: If *f*(*t*)
= ∑_{n=–∞}^{∞} *c*_{n}
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 L^{2}(\(\mathbb R\)^{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}∫\(\,_{{\mathbb R}^n}^~\) d^{n}*x* *f*(*x*)
exp{–i **k** · **x**}, and *f*(*x*) = (2π)^{–n/2}
∫\(\,_{{\mathbb R}^n}^~\) d^{n}*k* \(\tilde f\)(*k*) exp{i **k** · **x**}.

* __Examples__: For the Dirac delta
function,

δ^{(n)}(**x**–**x**_{0})
= (2π)^{–n} \(\int_{-\infty}^{+\infty}\)
d*k* exp{i **k** · (**x**–**x**_{0})}
;

For other common functions,

∫\(\,_{{\mathbb R}^n}^~\) d^{3}*x* exp{–i **k** · **x**} *r*^{–1} =
4π/*k*^{2} , ∫\(\,_{{\mathbb R}^n}^~\) d^{3}*k* exp{i **k** · **x**} *k*^{–2} =
2π^{2}/*r* ;

For a Gaussian *f*(*x*) = *N* exp{–*x*^{2}/2*σ*^{2}},
the Fourier transform \(\tilde f\)(*k*)
= *N'* exp{–*k*^{2}*σ*^{2}/2}
[the proof for *σ* = 1
is that the function *f*(*x*) = *N* exp{–*x*^{2}/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

*y*_{j} = *N*^{–1/2}
∑_{k}* x*_{k}
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
L^{2}(\(\mathbb R\),d*x*) 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 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)].

@ __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

∫ d*x* *f*(*x*) *g**(*x*)
= ∫ d*k* *F*(*k*) *G**(*k*) .

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