Circular summability 𝐶 of double trigonometric series

Abstract

and o-0)(x, y) are related in the following manner: o(n)(x, y) -L(x, y) =o(1) if and only if a()(x, y) -L(x, y) = o(1). This equivalence will be used in ?4. We shall say that the double trigonometric series T is circularly summable C at the point (x, y) if there exists an q such that the series is circular summable (C, 17) to a finite value at that point. The purpose of this paper is to do for double trigonometric series that which Plessner, see Zygmund [7, pp. 256-261], has done for single series, namely to give a necessary and sufficient condition that a given double trigonometric series be circular summable C. This goal is achieved in Theorem 5. It will be apparent from the definitions and proofs to be given that with appropriate modifications the results of this paper could be extended to multiple trigonometric series.