# 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 [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)

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).