scholarly journals Absolutely convergent Fourier series and generalized Zygmund classes of functions

2009 ◽  
Vol 104 (1) ◽  
pp. 124
Author(s):  
Ferenc Móricz
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Fourier Coefficients ◽  
Sufficient Conditions ◽  
Modulus Of Smoothness ◽  
Slowly Varying Function ◽  
Continuous Periodic Function ◽  
Convergent Fourier Series ◽  
Period 2 ◽  
Order Of Magnitude

We investigate the order of magnitude of the modulus of smoothness of a function $f$ with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that $f$ belongs to one of the generalized Zygmund classes $(\mathrm{Zyg}(\alpha, L)$ and $(\mathrm{Zyg} (\alpha, 1/L)$, where $0\le \alpha\le 2$ and $L= L(x)$ is a positive, nondecreasing, slowly varying function and such that $L(x) \to \infty$ as $x\to \infty$. A continuous periodic function $f$ with period $2\pi$ is said to belong to the class $(\mathrm{Zyg} (\alpha, L)$ if 26740 |f(x+h) - 2f(x) + f(x-h)| \le C h^\alpha L\left(\frac{1}{h}\right)\qquad \text{for all $x\in \mathsf T$ and $h>0$}, 26740 where the constant $C$ does not depend on $x$ and $h$; and the class $(\mathrm{Zyg} (\alpha, 1/L)$ is defined analogously. The above sufficient conditions are also necessary in case the Fourier coefficients of $f$ are all nonnegative.

On the almost everywhere convergence of Fourier series

1967 ◽  
Vol 63 (3) ◽  
pp. 703-705 ◽  
Author(s):  
B. S. Yadav
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Modulus Of Continuity ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Modulus Of Smoothness ◽  
General Concept ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere ◽  
Image Position

Let f be a 2π-periodic function of the class L(−π,π). PutWe call, with Žuk(6), the quantity L(p)(h, f) the L-modulus of smoothness of order p of the function f. Žuk has recently obtained, in (5) and (6), generalizations of a number of classical results on the absolute convergence of Fourier series, as also on the order of Fourier coefficients by employing the concept of the L-modulus of smoothness which is obviously a more general concept than that of the modulus of continuity. It is the purpose of this note to prove a theorem on the almost everywhere convergence of Fourier series of f involving the concept of L(p)(h, f).

scholarly journals Fourier Series with Small Gaps

2006 ◽  
Vol 13 (3) ◽  
pp. 581-584
Author(s):  
Rajendra G. Vyas
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Fourier Coefficients ◽  
Lacunary Series ◽  
Lacunary Fourier Series

Abstract Let 𝑓 be a 2π-periodic function in 𝐿1[–π, π] and be its lacunary Fourier series with small gaps. We have estimated Fourier coefficients of 𝑓 if it is of φ∧ 𝐵𝑉 locally. We have also obtained a precise interconnection between the lacunarity in such series and the localness of the hypothesis to be satisfied by the generic function which allows us to the interpolate the results concerning lacunary series and non-lacunary series.

Summability of general orthonormal Fourier series

2015 ◽  
Vol 52 (4) ◽  
pp. 511-536
Author(s):  
L. Gogoladze ◽  
V. Tsagareishvili
Keyword(s):  
Fourier Series ◽  
Fourier Coefficients ◽  
Orthonormal System ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

S. Banach in [1] proved that for any function f ∈ L2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α > 0. D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α > 0. In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φn(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.

On the Absolute Cesaro Summability of Negative Order of a Series Associated with the Conjugate Series of a Fourier Series

1969 ◽  
Vol 21 ◽  
pp. 552-557
Author(s):  
R. Mohanty ◽  
B. K. Ray
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Cesàro Summability ◽  
Conjugate Series ◽  
Cesaro Summability ◽  
Period 2 ◽  
Negative Order ◽  
Fixed Positive Number ◽  
The Absolute ◽  
Image Position

1. Definition. Let λ ≡ λ(ω) be continuous, differentiable, and monotonie increasing in (0, ∞) and let it tend to infinity as ω → ∞. A series an is summable |R, λ, r|, where r > 0, ifwhere A is a fixed positive number (6, Definition B).Let f(t) be a periodic function with period 2π and Lebesgue integrable over (–π, π) and let1.1The series conjugate to (1.1), at t = x, is1.2

scholarly journals On the absolute convergence of Fourier series and generalisation of Lipschitz spaces, defined by differences of fractional order

2016 ◽  
Vol 24 ◽  
pp. 77
Author(s):  
B.I. Peleshenko ◽  
T.N. Semirenko
Keyword(s):  
Fourier Series ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Sufficient Conditions ◽  
Periodic Functions ◽  
Lipschitz Spaces ◽  
Lipschitz Classes ◽  
Necessary And Sufficient

We obtain the necessary and sufficient conditions in terms of Fourier coefficients of $2\pi$-periodic functions $f$ with absolutely convergent Fourier series, for $f$ to belong to the generalized Lipschitz classes $H^{\omega, \alpha}_{\mathbb{C}}$, and to have the fractional derivative of order $\alpha$ ($0 < \alpha < 1$).

On the Absolute Convergence of Fourier Series

1997 ◽  
Vol 4 (4) ◽  
pp. 333-340
Author(s):  
T. Karchava
Keyword(s):  
Fourier Series ◽  
Finite Number ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Sufficient Conditions ◽  
Trigonometric Fourier Series ◽  
Periodic Functions ◽  
Continuity Modulus ◽  
The Absolute ◽  
Necessary And Sufficient

Abstract The necessary and sufficient conditions of the absolute convergence of a trigonometric Fourier series are established for continuous 2π-periodic functions which in [0, 2π] have a finite number of intervals of convexity, and whose 𝑛th Fourier coefficients are O(ω(1/𝑛; 𝑓)/𝑛), where ω(δ; 𝑓) is the continuity modulus of the function 𝑓.

On the Absolute Convergence of Fourier Series of Functions of ∧𝐵𝑉(𝑝) and φ ∧𝐵𝑉

2007 ◽  
Vol 14 (4) ◽  
pp. 769-774
Author(s):  
Rajendra G. Vyas
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Classical Result ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Natural Numbers ◽  
The Absolute ◽  
Series Of Functions ◽  
Sufficiency Conditions

Abstract Let 𝑓 be a 2π periodic function in 𝐿1[0,2π] and , be its Fourier coefficients. Extending the classical result of Zygmund, Schramm and Waterman obtained the sufficiency conditions for the absolute convergence of Fourier series of functions of ∧𝐵𝑉(𝑝) and φ ∧𝐵𝑉. Here we have generalized these results by obtaining certain sufficiency conditions for the convergence of the series , where is a strictly increasing sequence of natural numbers and 𝑛–𝑘 = –𝑛𝑘 for all 𝑘, for such functions.

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