Here,
g
{\displaystyle g}
is an arbitrary tempered distribution (e. . g. 1
A uniqueness theorem (or its proof) is, at least within the mathematics of differential equations, often combined with an existence theorem (or its proof) to a combined existence and uniqueness theorem (e. g. More generally, convolution in one domain (e.
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60 (2. Consider two sequences
g
[
n
]
{\displaystyle g[n]}
and
h
[
n
]
{\displaystyle h[n]}
with transforms
G
{\displaystyle G}
and
H
visit their website {\displaystyle H}
:
The §Discrete convolution of
g
{\displaystyle g}
and
h
{\displaystyle h}
is defined by:
The convolution theorem for discrete sequences is:34p.
The convolution theorem states that:12eq.
Consider functions
link g
,
h
{\displaystyle g,h}
in Lp-space
L
1
(
R
n
)
{\displaystyle L^{1}(\mathbb {R} ^{n})}
, with Fourier transforms
G
,
H
{\displaystyle G,H}
:
The convolution of
g
{\displaystyle g}
and
h
{\displaystyle h}
is defined by:
Also:
Hence by Fubini’s theorem we have that
r
L
1
(
R
n
)
{\displaystyle r\in L^{1}(\mathbb {R} ^{n})}
so its Fourier transform
R
{\displaystyle R}
is defined by the integral formula:
Note that
|
g
(
)
h
(
x
)
e
i
2
f
x
|
=
|
)
h
(
x
)
|
{\displaystyle |g(\tau )h(x-\tau )e^{-i2\pi f\cdot x}|=|g(\tau )h(x-\tau )|}
and hence by the argument above we may apply Fubini’s theorem again (i. .