ON THE INTEGRABILITY AND L(1)-CONVERGENCE OF COMPLEX TRIGONOMETRIC SERIES

We prove that if a weakly even sequence {c(k): k = 0, +/- 1, ...} of complex numbers is such that for some p > 1 we have [GRAPHICS] then the symmetric partial sums of the trigonometric series (*) SIGMA-k infinity = -infinity c(k)e(ikx) converge pointwise, except possibly at x = 0 (mod 2-pi), to a Lebesgue integrable function, (*) is the Fourier series of its sum, and series (*) converges in L1(-pi, pi)-norm if and only if lim\k\ --> infinity c(k) ln\k\ = 0. In addition, we present new proofs of the theorems by J. Fournier and W. Self [6] and by C. V. Stanojevic and V. B. Stanojevic [10].