ABSTRACT. The present paper proposes a new condition to replace both the (O-regularly varying) quasimonotone condition and a certain type of bounded variation condition, and shows the same conclusion for the uniform convergence of certain trigonometric series still holds. Theorem CJ. Suppose that {b n } is a non-increasing real sequence with lim n→∞ b n = 0. Then a necessary and sufficient condition for the uniform convergence of the seriesis lim n→∞ nb n = 0. Recently, Leindler [2] considered to generalize the monotonicity condition to a certain type of bounded variation condition and proved the following Theorem L. Let b= {b n } ∞ n=1 be a nonnegative sequence satisfying lim n→∞ b n = 0 andfor some constant M(b) depending only upon b and m = 1, 2, · · ·. Then a necessary and sufficient condition either for the uniform convergence of series (1), or for the continuity of its sum function f (x), is that lim n→∞ nb n = 0.

In the present paper, we give a brief review of L 1 -convergence of trigonometric series. Previous known results in this direction are improved and generalized by establishing a new condition.

We give an essential generalization of Telyakovskii's results on the convergence of trigonometric series with rarely changing coefficients in the norm of L 1 .

ABSTRACT. To verify the universal validity of the "two-sided" monotonicity condition introduced in [8], we will apply it to include more classical examples. The present paper selects the L p convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial.1991 Mathematics Subject Classification. 42A20 42A32In Fourier analysis, since Fourier coefficients are computable and applicable, people have established many nice results by assuming monotonicity of the coefficients. Generally speaking, it became an important topic how to generalize monotonicity. In many studies the generalization follows by this way (see, for example, [8] for definitions):(coefficients) nonincreasing ⇒ quasimonotone ⇒ regularly varying quasimonotoneOn the other hand, some mathematicians such as Leindler introduced "rest bounded variation" condition which also generalizes monotonicity: a nonnegative sequence {b n } with b n → 0 as n → ∞ is called of "rest bounded variation" (in symbol,for some constant M(b) depending only upon b and m = 1, 2, · · ·.Since quasimonotonicity and "rest bounded variation" are not comparable (cf. [6, Theorem 1]), we suggested the following condition (see [8]) to include both:Definition. Let c= {c n } ∞ n=1 be a nonnegative sequence tending to zero. If We can verify that either {b n } is (O-regularly varying) quasimonotone or {b n } ∈ R + 0 BV does imply that {b n } ∈ GBV (Zhou and Le [8, Theorem 3]). The converse is

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