Asymptotic Behavior of Trigonometric Series with O-Regularly Varying Quasimonotone Coefficients1

In this paper, the asymptotic behavior of a trigonometric series with O-regularly varying quasimonotone coefficients is investigated. Our results generalize the work done by S. Aljanˇci´c, R. Bojani´c, and M. Tomi´cŽ1956, Publ. Inst. Math.ŽBeograd. ŽN.S..10, 101 120., R. Bojani´c and J. KaramataŽ1963, Tech Rep 432, Math Research Center, Madison, WI., G. H. HardyŽ1928, J. London Math Soc. 3, 12 13; 1931, Proc. London Math. Soc. (2) 32, 441 448., G. H. Hardy and W. W. RogosinskiŽ1945, Quart. J. Math. Oxford 16, 49 58., J. R. Nurcombe Ž1993, J. Math. Anal. Appl. 178, 63 69., C. H. YongŽ1969, Publ. Inst. Math. Ž Beograd. ŽN.S..9, 29 35., and A. ZygmundŽ1968, ‘‘Trigonometric Series,’’ Vol. 1, 2nd ed., Cambridge Univ. Press, Cambridge, UK..