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 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

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

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