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A method for solving functional equations of the type:
 
A method for solving functional equations of the type:
  
<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/w/w097/w097910/w0979101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
A ( \lambda ) \Phi _ {+} ( \lambda ) + B ( \lambda ) \Phi _ {-} ( \lambda ) + C ( \lambda )  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w0979102.png" /> are given functions of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w0979103.png" />, analytic in a strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w0979104.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w0979105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w0979106.png" /> are non-zero in this strip. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w0979107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w0979108.png" /> are unknown functions of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w0979109.png" /> which tend to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791010.png" /> and are to be determined, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791011.png" /> being analytic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791013.png" /> being analytic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791014.png" />. Equation (1) must be satisfied in the entire strip of analyticity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791015.png" />.
+
where $  A ( \lambda ), B ( \lambda ), C ( \lambda ) $
 +
are given functions of a complex variable $  \lambda $,  
 +
analytic in a strip $  \tau _ {-} < \mathop{\rm Im}  \lambda < \tau _ {+} $,  
 +
and $  A ( \lambda ) $
 +
and $  B ( \lambda ) $
 +
are non-zero in this strip. The functions $  \Phi _ {+} ( \lambda ) $
 +
and $  \Phi _ {-} ( \lambda ) $
 +
are unknown functions of the complex variable $  \lambda $
 +
which tend to zero as $  | \lambda | \rightarrow \infty $
 +
and are to be determined, $  \Phi _ {+} ( \lambda ) $
 +
being analytic for $  \mathop{\rm Im}  \lambda > \tau _ {-} $
 +
and $  \Phi _ {-} ( \lambda ) $
 +
being analytic for $  \mathop{\rm Im}  \lambda < \tau _ {+} $.  
 +
Equation (1) must be satisfied in the entire strip of analyticity $  \tau _ {-} < \mathop{\rm Im}  \lambda < \tau _ {+} $.
  
 
The Wiener–Hopf method is based on the following two theorems.
 
The Wiener–Hopf method is based on the following two theorems.
  
1) A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791016.png" /> which is analytic in the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791017.png" /> and uniformly tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791018.png" /> can be represented inside this strip as a sum
+
1) A function $  F ( \lambda ) $
 +
which is analytic in the strip $  \tau _ {-} < \mathop{\rm Im}  \lambda < \tau _ {+} $
 +
and uniformly tends to zero as $  | \lambda | \rightarrow \infty $
 +
can be represented inside this strip as a sum
 +
 
 +
$$
 +
F ( \lambda )  =  F _ {+} ( \lambda ) + F _ {-} ( \lambda ),
 +
$$
 +
 
 +
where  $  F _ {+} ( \lambda ) $
 +
is analytic in the half-plane  $  \mathop{\rm Im}  \lambda > \tau _ {-} $,
 +
while  $  F _ {-} ( \lambda ) $
 +
is analytic in the half-plane  $  \mathop{\rm Im}  \lambda < \tau _ {+} $.
 +
 
 +
2) A function  $  F ( \lambda ) $
 +
which is analytic and non-zero in the strip  $  \tau _ {-} <  \mathop{\rm Im}  \lambda < \tau _ {+} $
 +
and which uniformly tends to one in this strip as  $  | \lambda | \rightarrow \infty $
 +
is representable in the given strip as a product:
  
<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/w/w097/w097910/w09791019.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
F ( \lambda )  = F _ {+} ( \lambda ) \cdot F _ {-} ( \lambda ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791020.png" /> is analytic in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791021.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791022.png" /> is analytic in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791023.png" />.
+
where $  F _ {+} ( \lambda ) $
 +
and  $  F _ {-} ( \lambda ) $
 +
are analytic and non-zero in the half-planes  $  \mathop{\rm Im}  \lambda > \tau _ {-} $
 +
and  $  \mathop{\rm Im}  \lambda < \tau _ {+} $,  
 +
respectively. The representation (2) is often called a factorization of the function  $  F ( \lambda ) $.
  
2) A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791024.png" /> which is analytic and non-zero in the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791025.png" /> and which uniformly tends to one in this strip as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791026.png" /> is representable in the given strip as a product:
+
The fundamental idea of the Wiener–Hopf method is that it is possible to factorize the function  $  L( \lambda ) = A ( \lambda ) /B( \lambda ) $;
 +
in other words, the method is based on the assumption that a representation
  
<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/w/w097/w097910/w09791027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{3 }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791029.png" /> are analytic and non-zero in the half-planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791031.png" />, respectively. The representation (2) is often called a factorization of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791032.png" />.
+
\frac{A ( \lambda ) }{B ( \lambda ) }
 +
  = \
  
The fundamental idea of the Wiener–Hopf method is that it is possible to factorize the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791033.png" />; in other words, the method is based on the assumption that a representation
+
\frac{L _ {+} ( \lambda ) }{L _ {-} ( \lambda ) }
  
<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/w/w097/w097910/w09791034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$
  
 
is possible. Using (3), equation (1) may be written as:
 
is possible. Using (3), equation (1) may be written 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/w/w097/w097910/w09791035.png" /></td> </tr></table>
+
$$
 +
L _ {+} ( \lambda ) \Phi _ {+} ( \lambda ) + L _ {-} ( \lambda ) \Phi _ {-} ( \lambda ) + L _ {-} ( \lambda )
 +
 
 +
\frac{C ( \lambda ) }{B ( \lambda ) }
 +
  = 0.
 +
$$
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791036.png" /> is analytic inside the strip, one has
+
Since $  L _ {-} ( \lambda ) C ( \lambda ) / B ( \lambda ) $
 +
is analytic inside the strip, one has
  
<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/w/w097/w097910/w09791037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
L _ {-} ( \lambda )
 +
\frac{C ( \lambda ) }{B ( \lambda ) }
 +
  = \
 +
D _ {+} ( \lambda ) + D _ {-} ( \lambda ).
 +
$$
  
 
Using (4), one finally obtains equation (1) in the form
 
Using (4), one finally obtains equation (1) 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/w/w097/w097910/w09791038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
L _ {+} \Phi _ {+} + D _ {+}  = - D _ {-} - L _ {-} \Phi _ {-} .
 +
$$
  
The left-hand side of (5) represents a function which is analytic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791039.png" />, while the right-hand side is a function which is analytic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791040.png" />. Since they have a common strip of analyticity in which condition (5) is satisfied, there exists a unique entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791041.png" /> which is identical with the left-hand and right-hand sides of (5), respectively, in their domains of analyticity. Hence
+
The left-hand side of (5) represents a function which is analytic for $  \mathop{\rm Im}  \lambda > \tau _ {-} $,  
 +
while the right-hand side is a function which is analytic for $  \mathop{\rm Im}  \lambda < \tau _ {+} $.  
 +
Since they have a common strip of analyticity in which condition (5) is satisfied, there exists a unique entire function $  P ( \lambda ) $
 +
which is identical with the left-hand and right-hand sides of (5), respectively, in their domains of analyticity. 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/w/w097/w097910/w09791042.png" /></td> </tr></table>
+
$$
 +
\Phi _ {+} ( \lambda )  = \
  
i.e. the solution of (1) is unique up to an entire function. If the order of growth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791044.png" /> is bounded at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791045.png" /> must be a polynomial. The functions sought are then determined uniquely up to constants, which are calculated by imposing additional conditions.
+
\frac{P ( \lambda ) - D _ {+} ( \lambda ) }{L _ {+} ( \lambda ) }
 +
,\ \
 +
\Phi _ {-} ( \lambda )  = \
 +
 
 +
\frac{- P ( \lambda ) - D _ {-} ( \lambda ) }{L _ {-} ( \lambda ) }
 +
,
 +
$$
 +
 
 +
i.e. the solution of (1) is unique up to an entire function. If the order of growth of $  L ( \lambda ) $
 +
and $  D ( \lambda ) $
 +
is bounded at infinity, $  P ( \lambda ) $
 +
must be a polynomial. The functions sought are then determined uniquely up to constants, which are calculated by imposing additional conditions.
  
 
The Wiener–Hopf method was developed in [[#References|[1]]] to solve special type integral equations (cf. [[Wiener–Hopf equation|Wiener–Hopf equation]]). It subsequently found extensive use in various problems of mathematical physics [[#References|[2]]].
 
The Wiener–Hopf method was developed in [[#References|[1]]] to solve special type integral equations (cf. [[Wiener–Hopf equation|Wiener–Hopf equation]]). It subsequently found extensive use in various problems of mathematical physics [[#References|[2]]].
Line 45: Line 125:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Wiener,  E. Hopf,  "Ueber eine Klasse singulärer Integralgleichungen"  ''Sitzungber. Akad. Wiss. Berlin''  (1931)  pp. 696–706</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Noble,  "Methods based on the Wiener–Hopf technique for the solution of partial differential equations" , Pergamon  (1958)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Wiener,  E. Hopf,  "Ueber eine Klasse singulärer Integralgleichungen"  ''Sitzungber. Akad. Wiss. Berlin''  (1931)  pp. 696–706</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Noble,  "Methods based on the Wiener–Hopf technique for the solution of partial differential equations" , Pergamon  (1958)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Theorem 2) as stated above is wrong; it requires an additional condition, namely that the winding number of the curve parametrized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791047.png" /> runs over the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791048.png" /> in the given strip, is equal to zero. So the Wiener–Hopf method described above works only under the additional requirement that the winding number condition is met for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791049.png" />. A detailed analysis of the Wiener–Hopf method for various classes of functions (not necessarily analytic on a strip) may be found in [[#References|[a1]]]. The matrix-valued version of this theory, which is due to [[#References|[a2]]] (see also [[#References|[a3]]]), is more complicated and explicit solutions can only obtained in special cases. The case when the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791051.png" /> appearing in equation (1) are rational matrix functions is of special interest and can be solved explicitly by employing a state-space method which is connected to mathematical systems theory (see [[#References|[a4]]], [[#References|[a5]]] and [[Integral equation of convolution type|Integral equation of convolution type]]).
+
Theorem 2) as stated above is wrong; it requires an additional condition, namely that the winding number of the curve parametrized by $  \lambda \mapsto F( \lambda ) $,  
 +
where $  \lambda $
 +
runs over the line $  \mathop{\rm Im}  \lambda = \tau $
 +
in the given strip, is equal to zero. So the Wiener–Hopf method described above works only under the additional requirement that the winding number condition is met for $  A ( \lambda ) / B ( \lambda ) $.  
 +
A detailed analysis of the Wiener–Hopf method for various classes of functions (not necessarily analytic on a strip) may be found in [[#References|[a1]]]. The matrix-valued version of this theory, which is due to [[#References|[a2]]] (see also [[#References|[a3]]]), is more complicated and explicit solutions can only obtained in special cases. The case when the functions $  A ( \lambda ) $
 +
and $  B ( \lambda ) $
 +
appearing in equation (1) are rational matrix functions is of special interest and can be solved explicitly by employing a state-space method which is connected to mathematical systems theory (see [[#References|[a4]]], [[#References|[a5]]] and [[Integral equation of convolution type|Integral equation of convolution type]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.G. Krein,  "Integral equations on a half-line with kernel depending upon the difference of the arguments"  ''Transl. Amer. Math. Soc. (2)'' , '''22'''  (1962)  pp. 163–288  ''Uspekhi Mat. Nauk'' , '''13''' :  5  (1958)  pp. 3–120</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.Ts. Gokhberg,  M.G. Krein,  "Systems of integral equations on a half line with kernels depending on the difference of arguments"  ''Transl. Amer. Math. Soc. (2)'' , '''14'''  (1960)  pp. 217–287  ''Uspekhi Mat. Nauk'' , '''13''' :  2 (80)  (1958)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.C. [I.Ts. Gokhberg] Gohberg,  I.A. Feld'man,  "Convolution equations and projection methods for their solution" , ''Transl. Math. Monogr.'' , '''41''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Bart,  I. Gohberg,  M.A. Kaashoek,  "Minimal factorization of matrix and operation functions" , Birkhäuser  (1979)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Gohberg,  M.A. Kaashoek,  "The state space method for solving singular integral equations"  A.C. Antoulas (ed.) , ''Mathematical System Theory. The influence of Kalman'' , Springer  (1991)  pp. 509–523</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Hochstadt,  "Integral equations" , Wiley  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.G. Krein,  "Integral equations on a half-line with kernel depending upon the difference of the arguments"  ''Transl. Amer. Math. Soc. (2)'' , '''22'''  (1962)  pp. 163–288  ''Uspekhi Mat. Nauk'' , '''13''' :  5  (1958)  pp. 3–120</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.Ts. Gokhberg,  M.G. Krein,  "Systems of integral equations on a half line with kernels depending on the difference of arguments"  ''Transl. Amer. Math. Soc. (2)'' , '''14'''  (1960)  pp. 217–287  ''Uspekhi Mat. Nauk'' , '''13''' :  2 (80)  (1958)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.C. [I.Ts. Gokhberg] Gohberg,  I.A. Feld'man,  "Convolution equations and projection methods for their solution" , ''Transl. Math. Monogr.'' , '''41''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Bart,  I. Gohberg,  M.A. Kaashoek,  "Minimal factorization of matrix and operation functions" , Birkhäuser  (1979)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Gohberg,  M.A. Kaashoek,  "The state space method for solving singular integral equations"  A.C. Antoulas (ed.) , ''Mathematical System Theory. The influence of Kalman'' , Springer  (1991)  pp. 509–523</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Hochstadt,  "Integral equations" , Wiley  (1973)</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


A method for solving functional equations of the type:

$$ \tag{1 } A ( \lambda ) \Phi _ {+} ( \lambda ) + B ( \lambda ) \Phi _ {-} ( \lambda ) + C ( \lambda ) = 0, $$

where $ A ( \lambda ), B ( \lambda ), C ( \lambda ) $ are given functions of a complex variable $ \lambda $, analytic in a strip $ \tau _ {-} < \mathop{\rm Im} \lambda < \tau _ {+} $, and $ A ( \lambda ) $ and $ B ( \lambda ) $ are non-zero in this strip. The functions $ \Phi _ {+} ( \lambda ) $ and $ \Phi _ {-} ( \lambda ) $ are unknown functions of the complex variable $ \lambda $ which tend to zero as $ | \lambda | \rightarrow \infty $ and are to be determined, $ \Phi _ {+} ( \lambda ) $ being analytic for $ \mathop{\rm Im} \lambda > \tau _ {-} $ and $ \Phi _ {-} ( \lambda ) $ being analytic for $ \mathop{\rm Im} \lambda < \tau _ {+} $. Equation (1) must be satisfied in the entire strip of analyticity $ \tau _ {-} < \mathop{\rm Im} \lambda < \tau _ {+} $.

The Wiener–Hopf method is based on the following two theorems.

1) A function $ F ( \lambda ) $ which is analytic in the strip $ \tau _ {-} < \mathop{\rm Im} \lambda < \tau _ {+} $ and uniformly tends to zero as $ | \lambda | \rightarrow \infty $ can be represented inside this strip as a sum

$$ F ( \lambda ) = F _ {+} ( \lambda ) + F _ {-} ( \lambda ), $$

where $ F _ {+} ( \lambda ) $ is analytic in the half-plane $ \mathop{\rm Im} \lambda > \tau _ {-} $, while $ F _ {-} ( \lambda ) $ is analytic in the half-plane $ \mathop{\rm Im} \lambda < \tau _ {+} $.

2) A function $ F ( \lambda ) $ which is analytic and non-zero in the strip $ \tau _ {-} < \mathop{\rm Im} \lambda < \tau _ {+} $ and which uniformly tends to one in this strip as $ | \lambda | \rightarrow \infty $ is representable in the given strip as a product:

$$ \tag{2 } F ( \lambda ) = F _ {+} ( \lambda ) \cdot F _ {-} ( \lambda ), $$

where $ F _ {+} ( \lambda ) $ and $ F _ {-} ( \lambda ) $ are analytic and non-zero in the half-planes $ \mathop{\rm Im} \lambda > \tau _ {-} $ and $ \mathop{\rm Im} \lambda < \tau _ {+} $, respectively. The representation (2) is often called a factorization of the function $ F ( \lambda ) $.

The fundamental idea of the Wiener–Hopf method is that it is possible to factorize the function $ L( \lambda ) = A ( \lambda ) /B( \lambda ) $; in other words, the method is based on the assumption that a representation

$$ \tag{3 } \frac{A ( \lambda ) }{B ( \lambda ) } = \ \frac{L _ {+} ( \lambda ) }{L _ {-} ( \lambda ) } $$

is possible. Using (3), equation (1) may be written as:

$$ L _ {+} ( \lambda ) \Phi _ {+} ( \lambda ) + L _ {-} ( \lambda ) \Phi _ {-} ( \lambda ) + L _ {-} ( \lambda ) \frac{C ( \lambda ) }{B ( \lambda ) } = 0. $$

Since $ L _ {-} ( \lambda ) C ( \lambda ) / B ( \lambda ) $ is analytic inside the strip, one has

$$ \tag{4 } L _ {-} ( \lambda ) \frac{C ( \lambda ) }{B ( \lambda ) } = \ D _ {+} ( \lambda ) + D _ {-} ( \lambda ). $$

Using (4), one finally obtains equation (1) in the form

$$ \tag{5 } L _ {+} \Phi _ {+} + D _ {+} = - D _ {-} - L _ {-} \Phi _ {-} . $$

The left-hand side of (5) represents a function which is analytic for $ \mathop{\rm Im} \lambda > \tau _ {-} $, while the right-hand side is a function which is analytic for $ \mathop{\rm Im} \lambda < \tau _ {+} $. Since they have a common strip of analyticity in which condition (5) is satisfied, there exists a unique entire function $ P ( \lambda ) $ which is identical with the left-hand and right-hand sides of (5), respectively, in their domains of analyticity. Hence

$$ \Phi _ {+} ( \lambda ) = \ \frac{P ( \lambda ) - D _ {+} ( \lambda ) }{L _ {+} ( \lambda ) } ,\ \ \Phi _ {-} ( \lambda ) = \ \frac{- P ( \lambda ) - D _ {-} ( \lambda ) }{L _ {-} ( \lambda ) } , $$

i.e. the solution of (1) is unique up to an entire function. If the order of growth of $ L ( \lambda ) $ and $ D ( \lambda ) $ is bounded at infinity, $ P ( \lambda ) $ must be a polynomial. The functions sought are then determined uniquely up to constants, which are calculated by imposing additional conditions.

The Wiener–Hopf method was developed in [1] to solve special type integral equations (cf. Wiener–Hopf equation). It subsequently found extensive use in various problems of mathematical physics [2].

References

[1] N. Wiener, E. Hopf, "Ueber eine Klasse singulärer Integralgleichungen" Sitzungber. Akad. Wiss. Berlin (1931) pp. 696–706
[2] B. Noble, "Methods based on the Wiener–Hopf technique for the solution of partial differential equations" , Pergamon (1958)

Comments

Theorem 2) as stated above is wrong; it requires an additional condition, namely that the winding number of the curve parametrized by $ \lambda \mapsto F( \lambda ) $, where $ \lambda $ runs over the line $ \mathop{\rm Im} \lambda = \tau $ in the given strip, is equal to zero. So the Wiener–Hopf method described above works only under the additional requirement that the winding number condition is met for $ A ( \lambda ) / B ( \lambda ) $. A detailed analysis of the Wiener–Hopf method for various classes of functions (not necessarily analytic on a strip) may be found in [a1]. The matrix-valued version of this theory, which is due to [a2] (see also [a3]), is more complicated and explicit solutions can only obtained in special cases. The case when the functions $ A ( \lambda ) $ and $ B ( \lambda ) $ appearing in equation (1) are rational matrix functions is of special interest and can be solved explicitly by employing a state-space method which is connected to mathematical systems theory (see [a4], [a5] and Integral equation of convolution type).

References

[a1] M.G. Krein, "Integral equations on a half-line with kernel depending upon the difference of the arguments" Transl. Amer. Math. Soc. (2) , 22 (1962) pp. 163–288 Uspekhi Mat. Nauk , 13 : 5 (1958) pp. 3–120
[a2] I.Ts. Gokhberg, M.G. Krein, "Systems of integral equations on a half line with kernels depending on the difference of arguments" Transl. Amer. Math. Soc. (2) , 14 (1960) pp. 217–287 Uspekhi Mat. Nauk , 13 : 2 (80) (1958)
[a3] I.C. [I.Ts. Gokhberg] Gohberg, I.A. Feld'man, "Convolution equations and projection methods for their solution" , Transl. Math. Monogr. , 41 , Amer. Math. Soc. (1974) (Translated from Russian)
[a4] H. Bart, I. Gohberg, M.A. Kaashoek, "Minimal factorization of matrix and operation functions" , Birkhäuser (1979)
[a5] I. Gohberg, M.A. Kaashoek, "The state space method for solving singular integral equations" A.C. Antoulas (ed.) , Mathematical System Theory. The influence of Kalman , Springer (1991) pp. 509–523
[a6] H. Hochstadt, "Integral equations" , Wiley (1973)
How to Cite This Entry:
Wiener-Hopf method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener-Hopf_method&oldid=49216
This article was adapted from an original article by V.I. Dmitriev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article