View source for Wiener-Hopf method

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>

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" />.

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

<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>

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" />.

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:

<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>

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" />.

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

<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:

<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>

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097910/w09791036.png" /> 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>

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>

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

<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>

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.

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]]].

====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>



====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]]).

====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>