## On solutions of integral equations with analytic kernels and rotations

### Volume 63 / 1996

Annales Polonici Mathematici 63 (1996), 293-300
DOI: 10.4064/ap-63-3-293-300

#### Abstract

We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*) x(t) + a(t)(Tx)(t) = b(t), where $T = M_{n₁,k₁} ... M_{n_m,k_m}$ and $M_{n_j,k_j}$ are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial $P_T(t) = t³ - t$. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation (*) in closed form.