An analog of the Dirichlet-Jordan
theorem and a uniqueness theorem are established for dual trigonometric
series equations when the right hand sides of the dual equations are given
functions of bounded variation. In the usual fashion there are two series in these
equations one of which has coefficients, say, {j∕nn} or {jn∕n − 1∕2}, and the
other coefficients {jn}. In the first series we establish ordinary convergence
and in the second Abel-Poisson convergence. In general jn≠0(1) and the
second series does not converge in the ordinary sense on any set of positive
measure. A best possible estimate on growth conditions for {jn} needed for
uniqueness is given. In the proof a mixed boundary value problem of potential
theory is associated with the dual series. Conformal mapping replaces this
potential problem with one in which a Dirichlet boundary condition can be
associated with the dual series. Analysis of this new problem provides the
denouement.
