An approach to the approximate solution of singular integral equations with a non-Carleman shift
Assume that \(\alpha(t)\) is an orientation-preserving non-Carleman diffeomorphism of a closed path \(\Gamma\) onto itself; assume that this path is sufficiently smooth and the diffeomorphism has a finite set of fixed points \(M(\alpha, \Gamma)= \{\tau_1, \tau_2, \dots, \tau_l\}\), \(\alpha (\tau_j)= \tau_j\), \(l\geq 1\). In the space \(L_2 (\Gamma)\), consider the isometric shift operator \[ (U\varphi) (t)=\sqrt {\bigl|\alpha'(t)} \bigr|\varphi \bigl( \alpha(t) \bigr), \] the singular integration operator \[ (S\varphi)(t) ={1\over \pi i} \int_\Gamma \varphi (\tau) (\tau-t)^{-1} d\tau, \] the mutually complementary projectors \[ P_\pm ={1\over 2} (I \pm S) \] and the operator \[ K=A P_+ +BP_-,\tag{1} \] where \(A= al+bU\), \(B=cl +dU\), \(a,b,c,d \in C(\Gamma)\), and \(I\) is the identity operator. The problem of calculating or estimating defect numbers \(\dim\ker K\) and \(\dim \text{coker} K\) and, all the more, the problem of describing the defect subspaces \(\ker K\) and \(\text{coker} K\), and the problem of an explicit or approximate solution of the equation \[ K\varphi =f \tag{2} \] are extremely difficult even in the case of the Carleman shift. If the parameters are constant, one can reduce this problem to the solution of the Wiener-Hopf integral equation on the half-line \(\mathbb{R}_+ =[0, \infty)\) with the kernel \(k(t- \tau)\). For a nonstationary system that is characterized, e.g., by an oscillating parameter, one can reduce the problem of synthesis to the integral equation on \(\mathbb{R}\) with the kernel \(h(t,\tau)= k(t-\tau) e^{ia \tau}\); \(a\) is a constant. After this, one can reduce this equation to the equation \[ T\varphi= \bigl[(I- c(t)U\bigr) P_++P_-\bigr] \varphi= \psi, \] by means of the Fourier transform and a conformal mapping; here, \(U\) is a non-Carleman shift operator with a nonempty \(M(\alpha, \mathbb{T})\). The Fredholm property criterion implies that either \(|c(\tau_j) |<1\) for all \(\tau_j\in M(\alpha, \mathbb{T})\) or \(c(t)\neq 0\) on \(\mathbb{T}\) and \(|c(\tau_j) |> 1\), \(j=1, \dots, l\). In this work, we provide an estimate of the defect number \(\alpha= \dim \ker \mathbb{T}\) in the case \(|c(\tau_j) |<1\), \(j=1, \dots,l\), and indicate an algorithm for calculating this estimate approximately; if the kernel is trivial, we also suggest an algorithm of approximate solution of the nonhomogeneous equation.
scientific article; zbMATH DE number 1113107