Page 587 - Handbook Of Integral Equations

Remark 2. For functions with nontrivial growth at infinity, the complete solution of Wiener–
Hopf equations of the second kind is presented in the cited book by F. D. Gakhov and Yu. I. Cherskii
(1978).

Remark 3. The Wiener–Hopf method can be applied to solve Wiener–Hopf integral equations
of the first kind under the assumption that the kernels of these equations are even.

•
References for Section 11.10: B. Noble (1958), A. G. Sveshnikov and A. N. Tikhonov (1970), V. I. Smirnov (1974),
F. D. Gakhov (1977), F. D. Gakhov and Yu. I. Cherskii (1978).

11.11. Krein’s Method for Wiener–Hopf Equations

11.11-1. Some Remarks. The Factorization Problem
Consider the Wiener–Hopf equation of the second kind

∞

y(x) – K(x – t)y(t) dt = f(x), 0 ≤ x < ∞, (1)
0
where f(x), y(x) ∈ L 1 (0, ∞) and K(x) ∈ L 1 (–∞, ∞). Let us use the classes of functions that
can be represented as Fourier transforms (alternative Fourier transform in the asymmetric form, see
Subsection 7.4-3), of functions from L 1 (–∞, ∞), L 1 (0, ∞), and L 1 (–∞, 0). For brevity, instead of
these symbols we simply write L, L + , and L – . Let functions h(x), h 1 (x), and h 2 (x) belong to L, L + ,
and L – , respectively; in this case, their transforms can be represented in the form

0
∞ ∞
ˇ
ˇ
ˇ
H(u)= h(x)e iux dx, H 1 (u)= h 1 (x)e iux dx, H 2 (u)= h 2 (x)e iux dx.
–∞ 0 –∞
Let Q, Q + , and Q – be the classes of functions representable in the form
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
W(u)=1 + H(u), W 1 (u)=1 + H 1 (u), W 2 (u)=1 + H 2 (u), (2)
respectively, where the functions from the classes Q + and Q – , treated as functions of the complex
variable z = u + iv, are analytic for Im z > 0 and Im z < 0, respectively, and are continuous up to the
real axis.
ˇ
Let T(x) belong to L and let T (u) be its transform. Assume that
1 " $ ∞
ˇ
ˇ
ˇ
1 – T (u) ≠ 0, Ind[1 – T (u)] = arg[1 – T (u)] =0, –∞ < u < ∞. (3)
2π –∞
In this case there exists a q(x) ∈ L such that
∞

ˇ
ln[1 – T (u)] = q(x)e iux dx. (4)
–∞
ˇ
This formula readily implies the relation ln[1 – T (u)] → 0as u →±∞.
ˇ
In what follows, we apply the factorization of functions M(u) of the class Q that are continuous
ˇ
on the interval –∞ ≤ u ≤ ∞. Here the factorization means a representation of the function M(u)in
the form of a product
k
u – i
ˇ
ˇ
ˇ
M(u)= M + (u) M – (u), (5)
u + i
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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