Operator integration with an involution in the upper limit of integration (Q1048506)

In the paper under review, the operator \[ (Af)(x) = \int_{0}^{\theta(x)} f(t)\,dt, \quad x\in[0,1], \] is studied under the assumption that \(\theta\) is an involution of some type. The authors investigate the Fredholm resolvent \(R^\lambda=(I-\alpha A)^{-1}A\) (\(\lambda\in \mathbb C\)) of this operator. A sufficient condition for existence of \(R^\lambda\) in terms of some boundary value problem is obtained. The authors study the asymptotic behavior of \(R^\lambda\) at large \(|\lambda|\). The main theorem of the paper states some type of convergence of the partial sums of the Fourier expansion in the eigenfuncions of the operator under consideration.