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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 analyze the relationships of four classical and three recently defined classes of numerical sequences.
Abstract
The problem of approximation of continuous functions by Cesàro (C,α)-means, −1 < α < 0, in terms of L p and C-modulus of continuity is studied.
Abstract
The behaviour of the Cesàro means of trigonometric Fourier series of monotone type functions in the space of continuous functions is studied.
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
Abstract
Let f: R N → C be a periodic function with period 2π in each variable. We prove suffcient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series.
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.
