On L1 convergence of Fourier series of complex valued functions

2006

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

Section: ⇒ Regularly Varying Quasimonotone ⇒ O-regularly Varying Quasmentioning

confidence: 77%

A remark on ``two-sided'' monotonicity condition: an application to Lp convergence

2006

“…Theorem 5 and Theorem 6 were established forO-regularly varying quasimonotone sequences by Xie and Zhou [10], and for GBVS by Le and Zhou [4].…”

Section: Given a Trigonometric Seriesmentioning

confidence: 98%

A generalization of the monotonicity condition and applications

2007

“…First, except generalizing the coefficients from monotonicity to a wider condition, Logarithm Rest Bounded Variation condition, we will also drop the prior requirement f ∈ L 2π but directly consider the series (1) or (2).…”

Section: Then (5) Holds If and Only Ifmentioning

confidence: 99%

“…For those x where the trigonometric series converges, write f (x) := ∞ n=1 a n sin nx, (1) g(x) := ∞ n=1 a n cos nx, (2) and in general, let (3) h(x) := +∞ n=−∞ c n e inx .…”

Section: Introductionmentioning

confidence: 99%

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L 1-approximation rate of certain trigonometric series

2011

By employing new ideas and techniques, we will refigure out the whole frame of L 1 -approximation. First, except generalizing the coefficients from monotonicity to a wider condition, Logarithm Rest Bounded Variation condition, we will also drop the prior requirement f ∈ L2π but directly consider the sine or cosine series. Secondly, to achieve nontrivial generalizations in complex spaces, we use a one-sided condition with some kind of balance conditions. In addition, a conjecture raised in [9] is disproved in Section 3.