Solution of a functional equation and applications
The problem of the vibration of the cochlea leads to a functional equation fo the form \((*)\) \(f(x + i) = d(x) f(x - i)\), \(x \in\mathbb{R}\), where \(d(x)\) is a given function and \(f\), a function to be found, is analytic in the strip \(-1<y<1\) in the \(z\)-plane, \(z = x + iy\). In this paper the author shows how to solve equation \((*)\) for a large class of functions \(d\) by solving the Dirichlet problem for the Laplace equation in a strip. First the case that \(d(x)\) is positive and satisfies the growth condition \(| \ln d(x) | < M (1 + | x |)^ m\), \(M,m\) constants, is treated and then it is shown how to reduce the case of complex \(d(x)\) to the case when \(d\) is positive. The given solution method is applied to treat \(d\) given by \[ d(x) = \gamma_ 0 \cdot e^{\beta_ 0 i} \cdot | x | \cdot e^{-2 \pi \alpha (x)i}, \] where \(\gamma_ 0\), \(\beta_ 0\) are constants, \(0<\beta_ 0 < 2\pi\), and \(\alpha (x)\) is the characteristic function of the positive \(x\)-axis. Finally, the author shows how the given solution method can be used to construct explicitly a spectral representation for a class of integral operators reminiscent of Wiener-Hopf operators: \(Pv(t) = e^{-t} \int^ \infty_{- \infty} c(t - s) e^{-s} v(s)ds\), where the function \(c\) is the Fourier inverse of \(d\) in \((*)\).
integral operators reminiscent of Wiener-Hopf operators
scientific article; zbMATH DE number 599813