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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c0264302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c0264303.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c0264304.png" />''
+
{{TEX|done}}
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c0264305.png" /> defined 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/c0264306.png" /></td> </tr></table>
 
  
it is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c0264307.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c0264308.png" /> is defined almost everywhere and also belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c0264309.png" />. The convolution has the basic properties of multiplication, namely,
+
''$f$ and $g$ belonging to $L(-\infty, +\infty)$''
  
<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/c02643010.png" /></td> </tr></table>
+
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}
  
<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/c02643011.png" /></td> </tr></table>
+
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|Banach algebra]] (for this norm $\|f*g\|\leq \|f\|\cdot \|g\|$).  
  
<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/c02643012.png" /></td> </tr></table>
+
If $F[f]$ denotes the Fourier transform of $f$, then
  
for any three functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643013.png" />. Therefore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643014.png" /> 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
+
$$
 +
F [f * g] \  = \
 +
\sqrt {2 \pi}
 +
F [f] F [g] ,
 +
$$
  
<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/c02643015.png" /></td> </tr></table>
 
 
is a [[Banach algebra|Banach algebra]] (for this norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643016.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643017.png" /> denotes the Fourier transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026430/c02643018.png" />, 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>
 
  
 
and this is used in solving a number of applied problems.
 
and this is used in solving a number of applied problems.
Line 25: 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 <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,
+
where $  D $
 +
is the differentiation operator and $  \alpha $
 +
is any multi-index,
  
<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>
+
$$
 +
(D ^ \alpha  \delta ) * f \  = \
 +
D ^ \alpha  f,
 +
$$
  
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
 
  
<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>
+
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
  
then
+
$$
 +
K \  \supset \  \mathop{\rm supp}\nolimits \  f _{n} ,\ \
 +
n = 1,\  2 \dots
 +
$$
  
<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/c02643054.png" /></td> </tr></table>
 
  
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
+
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/c02643058.png" /></td> </tr></table>
+
$$
 +
f _{n} * g \  \rightarrow \
 +
f _{0} * g.
 +
$$
  
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
 
  
<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>
+
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
  
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.
+
$$
 +
F [f * g] \  = \
 +
\sqrt {2 \pi}
 +
F [f] F [g].
 +
$$
  
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.
 
  
====References====
+
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><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)  {{MR|0764399}} {{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}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  G.E. Shilov,  "Generalized functions" , '''1–5''' , Acad. Press  (1964)  (Translated from Russian)  {{MR|435831}} {{ZBL|0115.33101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.C. Titchmarsh,   "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)  {{MR|0942661}} {{ZBL|0017.40404}}  {{ZBL|63.0367.05}} </TD></TR></table>
 
  
 +
$$
 +
U (x,\  t) \  = \
 +
\mu (x) *
 +
{
 +
\frac{1}{2 \sqrt {\pi t}}
 +
}
 +
e ^ {-x ^{2} /4t} ,
 +
$$
  
  
====Comments====
+
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Kecs,  "The convolution product and some applications" , Reidel &amp; Ed. Academici  (1982)  {{MR|0690953}} {{ZBL|0512.46041}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)  {{MR|0764399}} {{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}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  G.E. Shilov,  "Generalized functions" , '''1–5''' , Acad. Press  (1964)  (Translated from Russian)  {{MR|435831}} {{ZBL|0115.33101}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  E.C. Titchmarsh,  "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)  {{MR|0942661}} {{ZBL|0017.40404}}  {{ZBL|63.0367.05}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Kecs,  "The convolution product and some applications" , Reidel &amp; Ed. Academici  (1982)  {{MR|0690953}} {{ZBL|0512.46041}} </TD></TR>
 +
</table>

Latest revision as of 19:44, 2 November 2023



$f$ 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
[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=28165
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article