INTEGRABILITY AND L-CONVERGENCE OF MULTIPLE TRIGONOMETRIC SERIES

It is proved that under the following condition, the sum f of the double trigonometric series with coefficients c(jk) is integrable and the rectangular partial sums s(mn)(f, x, y) converge to f in L1-norm: [GRAPHICS] This generalises the corresponding results of Moricz [3]. We also prove that the aforementioned condition is sharp. A more general version of this result is established for double series of orthonormal functions, which generalises Moricz-Schipp-Wade [4]. An extension to higher-dimensions is given.