Fourier
Analysis| Fourier
Analysis |
Fourier Series / Components of a Function
$ 1-dimensional: For f :
[–L, L] → C,
f(x) = (2L)–1/2 
k =
–infty+infty fk
exp{i
kx/L}, with fk:=
(2L)–1/2 
–LL dx f(x)
exp{–ikx/L}
.
$ n-dimensional:
For f :
Tn → C, i.e., 
i 
[0,
2],
obtained replacing x/L by in
the 1D case,
f()
= k in
Z^n fk exp{i k · }, with fk:=
(2)–n T^n dn f()
exp{–i
k · }
.
* Wiener's theorem: If f(t)
= n=–inftyinfty cn
exp{2i tn} is a
non-zero absolutely convergent (Fourier) series on [0,1], then
1/f(t)
can be represented as an absolutely convergent Fourier series
on this interval.
Fourier Transform of a Function > s.a. Convolution;
Gel'fand Transform; Wavelets [alternative].
* Idea: A map f 
(f)
= f ~ that gives the frequency distribution
in f; If f is
in L2(Rn),
then so is f ~.
$ Def: The Fourier transform of f and its inverse Fourier transform
are, respectively,
f ~(k):= (2)–n/2
R^n dnx f(x)
exp{–i k · x}, and f(x) = (2)–n/2
R^n dnk f ~(k) exp{i k · x}.
* Examples: For the Dirac delta function,
(n)(x–x0)
= (2)–n –infty+infty
dk exp{i k · (x–x0)}
;
For other common functions,
R^3 d3x exp{–i k · x} r–1 =
4/k2 , R^3 d3k exp{i k · x} k–2 =
22/r ;
For a Gaussian f(x) = N exp{–x2/2
2},
the Fourier transform f ~(k)
= N' exp{–k22/2}
[the proof for = 1 is
that the function f(x) = N exp{–x2/2}
is the general solution of the ode x f(x) + f '(x)
= 0, and the Fourier transform of that ode 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{2i
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(R,dx) for some y > 0
iff f ~(k) vanishes for k < 0,
and in that case f ~(k,y)
= exp{–ky} f ~(k).
@ General: Oberhettinger 73, 73 [tables]; Brigham 74; James 02 [II]; Bracewell
03 [and applications].
@ Special topics: Kempf JMP(00)gq/99 [finite
bandwidth & rapid variation]; Wurm et al JMP(03)
[of Lorentz-invariant functions]; Ozaktas et al 01 [fractional]; McCallum &
Horikis JPA(06) [self-Fourier functions].
Related Concepts and Results
$ 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
22 may 2008