Pointwise Convergence of Trigonometric Series

We establish two results in the pointwise convergence problem of a trigonometric series [An] £ cne inl with lim Hm £ I bTck | = 0 |n|< -x. * Jn-»oo \k\-n for some nonnegative integer m. These results not only generalize Hardy's theorem, the Jordan test theorem and Fatou's theorem, but also complement the results on pointwise convergence of those Fourier series associated with known 1}-convergence classes. A similar result is also established for the case that lim,,_,, £[y!"l""'|A"'c,,| = 0, where (/„} satisfies certain conditions. ft JU | n | •""" ft I A. I ' V ft j 1980 Mathematics subject classification (Amer. Math. Soc): 42 A 20, 42 A 32.