The conjugation problem on an infinite set of intervals

It is known that if path of integration consists of a finite number of intervals, then: (1) in the case of a Fredholm-type kernel, the index of the Fredholm operator is zero; (2) in the case of a Cauchy-type kernel, the index of the singular integration operator is a finite number (possible zero). Study of the conjugate boundary-value problem on an infinite set of intervals brings out new facts. The following may be noted: (1) A homogeneous boundary-value problem is always solvable in the classK, which is a natural generalization of that of piecewise analytic functions 1]. (2) Associated (conjugated) homogeneous boundary-value problems have any number of linearly independent solutions in the associated (conjugated) classes, so that the notion of class index is no longer relevant. (3) Associated (conjugated) homogeneous singular integral equations have any number of linearly independent solutions in the associated (conjugated) spacesL _p, L_q, p^?1+q^?1=1, so that the notion of operator index is no longer relevant The general theory of the problems under consideration is satisfactorily illustrated by the simplest case—a set of intervals on the real axis. For this reason the line of discontinuities (integration path) in the present paper is part of the real axis. The paper generalizes the results of 2–4]. Relevant work includes 5].