# Wiener-Hopf method

(Redirected from Wiener–Hopf method)

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  to solve special type integral equations (cf. Wiener–Hopf equation). It subsequently found extensive use in various problems of mathematical physics .

How to Cite This Entry:
Wiener–Hopf method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener%E2%80%93Hopf_method&oldid=23144