  Several Lebesgue-type decomposition theorems in analysis have a strong
relation to the operation called: parallel sum. The aim of this paper is to
investigate this relation from a new point of view. Namely, using a natural
generalization of Arlinskii's approach (which identifies the singular part as a
fixed point of a single-variable map) we prove the existence of a Lebesgue-type
decomposition for nonnegative sesquilinear forms. As applications, we also show
that how this approach can be used to derive analogous results for
representable functionals, nonnegative finitely additive measures, and positive
definite operator functions. The focus is on the fact that each theorem can be
proved with the same completely elementary method.
