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Difference between revisions of "Convolution of functions"

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 +
  
  
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If $F[f]$ denotes the Fourier transform of $f$, then
 
If $F[f]$ denotes the Fourier transform of $f$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643019.png" /></td> </tr></table>
+
$$
 +
F [f * g] \  = \
 +
\sqrt {2 \pi}
 +
F [f] F [g] ,
 +
$$
 +
 
  
 
and this is used in solving a number of applied problems.
 
and this is used in solving a number of applied problems.
Line 35: Line 42:
 
Thus, if a problem has been reduced to an integral equation of the form
 
Thus, if a problem has been reduced to an integral equation of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{*}
 +
f (x) \  = \  g (x) +
 +
\int\limits _ {- \infty} ^ \infty
 +
K (x - y) f (y) \  dy,
 +
$$
 +
 
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643021.png" /></td> </tr></table>
+
$$
 +
g (x) \  \in \
 +
L _{2} (- \infty ,\  \infty ),\ \
 +
K (x) \  \in \
 +
L (- \infty ,\  \infty ),
 +
$$
 +
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643022.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm sup} _{x} \  | F [K] (x) | \  \leq \ 
 +
\frac{1}{\sqrt {2 \pi}}
 +
,
 +
$$
  
then, under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643023.png" />, by applying the Fourier transformation to (*) one obtains
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643024.png" /></td> </tr></table>
+
then, under the assumption that  $  f \in L (- \infty ,\  \infty ) $,
 +
by applying the Fourier transformation to (*) one obtains
 +
 
 +
$$
 +
F [f] \  = \
 +
F [g] +
 +
\sqrt {2 \pi}
 +
F [f] F [K],
 +
$$
 +
 
  
 
hence
 
hence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643025.png" /></td> </tr></table>
+
$$
 +
F [f] \  = \
 +
 
 +
\frac{F [g]}{1 - \sqrt {2 \pi} F [K]}
 +
,
 +
$$
 +
 
  
 
and the inverse Fourier transformation yields the solution to (*) as
 
and the inverse Fourier transformation yields the solution to (*) as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643026.png" /></td> </tr></table>
+
$$
 +
f (x) \  = \
  
The properties of a convolution of functions have important applications in probability theory. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643028.png" /> are the probability densities of independent random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643030.png" />, respectively, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643031.png" /> is the probability density of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643032.png" />.
+
\frac{1}{\sqrt {2 \pi}}
  
The convolution operation can be extended to generalized functions (cf. [[Generalized function|Generalized function]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643034.png" /> are generalized functions such that at least one of them has compact support, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643035.png" /> is a test function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643036.png" /> is defined by
+
\int\limits _ {- \infty} ^ \infty
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643037.png" /></td> </tr></table>
+
\frac{F [g] ( \zeta ) e ^ {-i \zeta x}}{1 - \sqrt {2 \pi} F [K] ( \zeta )}
 +
\
 +
d \zeta .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643038.png" /> is the direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643040.png" />, that is, the functional on the space of test functions of two independent variables given by
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643041.png" /></td> </tr></table>
+
The properties of a convolution of functions have important applications in probability theory. If  $  f $
 +
and  $  g $
 +
are the probability densities of independent random variables  $  X $
 +
and  $  Y $,
 +
respectively, then  $  (f * g) $
 +
is the probability density of the random variable  $  X + Y $.
  
for every infinitely-differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643042.png" /> of compact support.
+
 
 +
The convolution operation can be extended to generalized functions (cf. [[Generalized function|Generalized function]]). If  $  f $
 +
and  $  g $
 +
are generalized functions such that at least one of them has compact support, and if  $  \phi $
 +
is a test function, then  $  f * g $
 +
is defined by
 +
 
 +
$$
 +
\langle  f * g,\  \phi \rangle \  = \
 +
\langle  f (x) \times g (y),\  \phi (x + y) \rangle,
 +
$$
 +
 
 +
 
 +
where  $  f (x) \times g (y) $
 +
is the direct product of  $  f $
 +
and  $  g $,
 +
that is, the functional on the space of test functions of two independent variables given by
 +
 
 +
$$
 +
\langle  f (x) \times g (y),\  u (x,\  y) \rangle \  = \
 +
< f (x),\  < g (y),\  u (x,\  y) \gg
 +
$$
 +
 
 +
 
 +
for every infinitely-differentiable function $  u (x,\  y) $
 +
of compact support.
  
 
The convolution of generalized functions also has the commutativity property and is linear in each argument; it is associative if at least two of the three generalized functions have compact supports. The following equalities hold:
 
The convolution of generalized functions also has the commutativity property and is linear in each argument; it is associative if at least two of the three generalized functions have compact supports. The following equalities hold:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643043.png" /></td> </tr></table>
+
$$
 +
D ^ \alpha  (f * g) \  = \
 +
D ^ \alpha  f * g \  = \
 +
f * D ^ \alpha  g,
 +
$$
 +
 
 +
 
 +
where  $  D $
 +
is the differentiation operator and  $  \alpha $
 +
is any multi-index,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643044.png" /> is the differentiation operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643045.png" /> is any multi-index,
+
$$
 +
(D ^ \alpha  \delta ) * f \  = \
 +
D ^ \alpha  f,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643046.png" /></td> </tr></table>
 
  
in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643047.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643048.png" /> denotes the delta-function. Also, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643050.png" /> are generalized functions such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643051.png" />, and if there is a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643052.png" /> such that
+
in particular, $  \delta * f = f $,  
 +
where $  \delta $
 +
denotes the delta-function. Also, if $  f _{n} $,
 +
$  n = 1,\  2 \dots $
 +
are generalized functions such that $  f _{n} \rightarrow f _{0} $,  
 +
and if there is a compact set $  K $
 +
such that
 +
 
 +
$$
 +
K \  \supset \  \mathop{\rm supp}\nolimits \  f _{n} ,\ \
 +
n = 1,\  2 \dots
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643053.png" /></td> </tr></table>
 
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643054.png" /></td> </tr></table>
+
$$
 +
f _{n} * g \  \rightarrow \
 +
f _{0} * g.
 +
$$
  
Finally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643055.png" /> is a generalized function of compact support and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643056.png" /> is a generalized function of slow growth, then the Fourier transformation can be applied to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643057.png" />, and again
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643058.png" /></td> </tr></table>
+
Finally, if  $  g $
 +
is a generalized function of compact support and  $  f $
 +
is a generalized function of slow growth, then the Fourier transformation can be applied to  $  f * g $,
 +
and again
 +
 
 +
$$
 +
F [f * g] \  = \
 +
\sqrt {2 \pi}
 +
F [f] F [g].
 +
$$
 +
 
  
 
The convolution of generalized functions is widely used in solving boundary value problems for partial differential equations. Thus, the Poisson integral, written in the form
 
The convolution of generalized functions is widely used in solving boundary value problems for partial differential equations. Thus, the Poisson integral, written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643059.png" /></td> </tr></table>
+
$$
 +
U (x,\  t) \  = \
 +
\mu (x) *
 +
{
 +
\frac{1}{2 \sqrt {\pi t}}
 +
}
 +
e ^ {-x ^{2} /4t} ,
 +
$$
 +
 
  
is a solution to the thermal-conductance equation for an infinite bar, where the initial temperature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643060.png" /> can be not only an ordinary function but also a generalized one.
+
is a solution to the thermal-conductance equation for an infinite bar, where the initial temperature $  \mu $
 +
can be not only an ordinary function but also a generalized one.
  
Both for ordinary and generalized functions the concept of a convolution carries over in a natural way to functions of several variables; then in the above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643062.png" /> must be regarded as vectors from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643063.png" /> and not as real numbers.
+
Both for ordinary and generalized functions the concept of a convolution carries over in a natural way to functions of several variables; then in the above $  x $
 +
and $  y $
 +
must be regarded as vectors from $  \mathbf R ^{n} $
 +
and not as real numbers.
  
 
====References====
 
====References====

Revision as of 16:36, 28 January 2020



$f$ and $g$ belonging to $L(-\infty, +\infty)$

The function $h$ defined by \begin{equation} h(x) = \int\limits_{-\infty}^{+\infty}f(x-y)g(y)\,dy = \int\limits_{-\infty}^{+\infty}f(y)g(x-y)\,dy; \end{equation} it is denoted by the symbol $f*g$. The function $f*g$ is defined almost everywhere and also belongs to $L(-\infty, +\infty)$.

Properties

The convolution has the basic properties of multiplication, namely, \begin{equation} f*g = g*f, \end{equation} \begin{equation} (\alpha_1f_1 + \alpha_2f_2)*g = \alpha_1(f_1*g) + \alpha_2(f_2*g), \quad \alpha_1, \alpha_2 \in \mathbb{R}, \end{equation} \begin{equation} (f*g)*h = f*(g*h) \end{equation}

for any three functions in $L(-\infty, \infty)$. Therefore, $L(-\infty, \infty)$ with the usual operations of addition and of multiplication by a scalar, with the operation of convolution as the multiplication of elements, and with the norm \begin{equation} \|f\| = \int\limits_{-\infty}^{\infty}|f(x)|\, dx, \end{equation} is a Banach algebra (for this norm $\|f*g\|\leq \|f\|\cdot \|g\|$).

If $F[f]$ denotes the Fourier transform of $f$, then

$$ F [f * g] \ = \ \sqrt {2 \pi} F [f] F [g] , $$


and this is used in solving a number of applied problems.

Thus, if a problem has been reduced to an integral equation of the form

$$ \tag{*} f (x) \ = \ g (x) + \int\limits _ {- \infty} ^ \infty K (x - y) f (y) \ dy, $$


where

$$ g (x) \ \in \ L _{2} (- \infty ,\ \infty ),\ \ K (x) \ \in \ L (- \infty ,\ \infty ), $$


$$ \mathop{\rm sup} _{x} \ | F [K] (x) | \ \leq \ \frac{1}{\sqrt {2 \pi}} , $$


then, under the assumption that $ f \in L (- \infty ,\ \infty ) $, by applying the Fourier transformation to (*) one obtains

$$ F [f] \ = \ F [g] + \sqrt {2 \pi} F [f] F [K], $$


hence

$$ F [f] \ = \ \frac{F [g]}{1 - \sqrt {2 \pi} F [K]} , $$


and the inverse Fourier transformation yields the solution to (*) as

$$ f (x) \ = \ \frac{1}{\sqrt {2 \pi}} \int\limits _ {- \infty} ^ \infty \frac{F [g] ( \zeta ) e ^ {-i \zeta x}}{1 - \sqrt {2 \pi} F [K] ( \zeta )} \ d \zeta . $$


The properties of a convolution of functions have important applications in probability theory. If $ f $ and $ g $ are the probability densities of independent random variables $ X $ and $ Y $, respectively, then $ (f * g) $ is the probability density of the random variable $ X + Y $.


The convolution operation can be extended to generalized functions (cf. Generalized function). If $ f $ and $ g $ are generalized functions such that at least one of them has compact support, and if $ \phi $ is a test function, then $ f * g $ is defined by

$$ \langle f * g,\ \phi \rangle \ = \ \langle f (x) \times g (y),\ \phi (x + y) \rangle, $$


where $ f (x) \times g (y) $ is the direct product of $ f $ and $ g $, that is, the functional on the space of test functions of two independent variables given by

$$ \langle f (x) \times g (y),\ u (x,\ y) \rangle \ = \ < f (x),\ < g (y),\ u (x,\ y) \gg $$


for every infinitely-differentiable function $ u (x,\ y) $ of compact support.

The convolution of generalized functions also has the commutativity property and is linear in each argument; it is associative if at least two of the three generalized functions have compact supports. The following equalities hold:

$$ D ^ \alpha (f * g) \ = \ D ^ \alpha f * g \ = \ f * D ^ \alpha g, $$


where $ D $ is the differentiation operator and $ \alpha $ is any multi-index,

$$ (D ^ \alpha \delta ) * f \ = \ D ^ \alpha f, $$


in particular, $ \delta * f = f $, where $ \delta $ denotes the delta-function. Also, if $ f _{n} $, $ n = 1,\ 2 \dots $ are generalized functions such that $ f _{n} \rightarrow f _{0} $, and if there is a compact set $ K $ such that

$$ K \ \supset \ \mathop{\rm supp}\nolimits \ f _{n} ,\ \ n = 1,\ 2 \dots $$


then

$$ f _{n} * g \ \rightarrow \ f _{0} * g. $$


Finally, if $ g $ is a generalized function of compact support and $ f $ is a generalized function of slow growth, then the Fourier transformation can be applied to $ f * g $, and again

$$ F [f * g] \ = \ \sqrt {2 \pi} F [f] F [g]. $$


The convolution of generalized functions is widely used in solving boundary value problems for partial differential equations. Thus, the Poisson integral, written in the form

$$ U (x,\ t) \ = \ \mu (x) * { \frac{1}{2 \sqrt {\pi t}} } e ^ {-x ^{2} /4t} , $$


is a solution to the thermal-conductance equation for an infinite bar, where the initial temperature $ \mu $ can be not only an ordinary function but also a generalized one.

Both for ordinary and generalized functions the concept of a convolution carries over in a natural way to functions of several variables; then in the above $ x $ and $ y $ must be regarded as vectors from $ \mathbf R ^{n} $ and not as real numbers.

References

[1] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) MR0764399 Zbl 0954.35001 Zbl 0652.35002 Zbl 0695.35001 Zbl 0699.35005 Zbl 0607.35001 Zbl 0506.35001 Zbl 0223.35002 Zbl 0231.35002 Zbl 0207.09101
[2] I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1–5 , Acad. Press (1964) (Translated from Russian) MR435831 Zbl 0115.33101
[3] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) MR0942661 Zbl 0017.40404 Zbl 63.0367.05


Comments

References

[a1] W. Kecs, "The convolution product and some applications" , Reidel & Ed. Academici (1982) MR0690953 Zbl 0512.46041
How to Cite This Entry:
Convolution of functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convolution_of_functions&oldid=29335
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article