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1. The Fourier transform on R. is integrable; this means, by definition, f Leb1. For each R we set
 

Summary: 1. The Fourier transform on R.
Suppose
f : R C
is integrable; this means, by definition, f Leb1. For each R we set
^f() = f(x)e-ix
dx.
Evidently,
(0) | ^f()| |f(x)| dx whenever x R.
This function is called the Fourier transform of f. Note that
(1) lim
||
^f() = 0
by the Riemann-Lebesgue Lemma.
Definition 1.1. Suppose f : R C. We let
spt f = cl {f = 0}
and call this closed set the support of f. We say f is smooth if dmn f(m)
= R
for all m N.
We let
D(R)

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics