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)

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)