Singular integral operators in a weighted \(L^ 2\)-space (Q1067590)

Cauchy singular integral operators are characterized as operators in a weighted \(L^ 2\)-space. The integral operator arises from a singular integral equation with variable coefficients. An appropriate weight function associated with the singular integral operator is constructed, and the set of polynomials orthogonal with respect to this weight function is defined. The action of the operator on polynomial sets is studied, and the definition of the operator is extended to a weighted \(L^ 2\)-space. In this space, the operator is shown to be bounded, and, in some cases, isometric. Formulas are developed for the composition of the singular integral operator and its one sided inverse.