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Abstract
Let f∊L
2π
be a real-valued even function with its Fourier series 
, and let S
n
(f,x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence {a
n
} is decreasing and 
, then 
if and only if 
. Later, the monotonicity condition set on {a
n
} is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM7 condition in real space. In this paper, we give a complete generalization to the complex space.
Abstract
We study L
p
-integrability (1<p<∞) of a sum ϕ of trigonometric series under the assumptions that the sequence of coefficients of ϕ belongs to the class 
. Then we discuss the relations between the properties of ϕ and the properties of the sequence (λ
n
)∊GM(β,r), and deduce an estimate for modulus of continuity of ϕ in L
p
norm.
Abstract
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∊L 2π 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.
Abstract
We introduce a new class of sequences called NBVS to generalize GBVS, essentially extending monotonicity from “one sided” to “two sided”, while some important classical results keep true.
Abstract
We study when sums of trigonometric series belong to given function classes. For this purpose we describe the Nikol’skii class of functions and, in particular, the generalized Lipschitz class. Results for series with positive and general monotone coefficients are presented.
Abstract
We introduce the higher order Lipschitz classes Λ r (α) and λ r (α) of periodic functions by means of the rth order difference operator, where r = 1, 2, ..., and 0 < α ≦ r. We study the smoothness property of a function f with absolutely convergent Fourier series and give best possible sufficient conditions in terms of its Fourier coefficients in order that f belongs to one of the above classes.
Abstract
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.
Abstract
Chaundry and Jolliffe [1] proved that if a k is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence of the series Σ k=1 ∞ a k sin kx is lim k→∞ ka k = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu [4] generalized this classical result. In this paper we propose new classes of sequences for which we get the extended version of their results. Moreover, we generalize the results of S. Tikhonov [2] on the L 1-convergence of Fourier series.
