Vol. 59, No. 1, 1975

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A Dirichlet-Jordan theorem for dual trigonometric series

Robert Bernard Kelman

Vol. 59 (1975), No. 1, 113–123

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 12}, and the other coefficients {jn}. In the first series we establish ordinary convergence and in the second Abel-Poisson convergence. In general jn0(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.

Mathematical Subject Classification 2000
Primary: 42A20
Secondary: 45F10
Received: 12 August 1973
Revised: 6 May 1974
Published: 1 July 1975
Robert Bernard Kelman