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.

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 generalized absolute convergence of Fourier series

2019 ◽  
Vol 26 (1) ◽  
pp. 117-124
Author(s):  
Rusudan Meskhia
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Absolute Convergence ◽  
Sufficient Conditions ◽  
Trigonometric Fourier Series ◽  
Fourier Trigonometric Series ◽  
Variation Of A Function ◽  
The Absolute

Abstract In the present paper the sufficient conditions are obtained for the generalized r-absolute convergence ( {0<r<2} ) of the single Fourier trigonometric series in terms of the modulus of δ-variation of a function. It is proved that these conditions are unimprovable in a certain sense. The classical results of Berstein, Szasz, Zygmund and others, related to the absolute convergence of single trigonometric Fourier series, were previously generalized by [L. Gogoladze and R. Meskhia, On the absolute convergence of trigonometric Fourier series, Proc. A. Razmadze Math. Inst. 141 2006, 29–40].

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.

scholarly journals Fourier transforms of Dini-Lipschitz functions

1986 ◽  
Vol 9 (2) ◽  
pp. 301-312 ◽  
Author(s):  
M. S. Younis
Keyword(s):  
Fourier Series ◽  
Dual Space ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Fourier Transforms ◽  
Function Class ◽  
Lipschitz Functions ◽  
Lipschitz Class ◽  
The Absolute ◽  
Lipschitz Conditions

It is well known that if Lipschitz conditions of a certain order are imposed on a functionf(x), then these conditions affect considerably the absolute convergence of the Fourier series and Fourier transforms off. In general, iff(x)belongs to a certain function class, then the Lipschitz conditions have bearing as to the dual space to which the Fourier coefficients and transforms off(x)belong. In the present work we do study the same phenomena for the wider Dini-Lipschitz class as well as for some other allied classes of functions.

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.

scholarly journals On linear independence for integer translates of a finite number of functions

1993 ◽  
Vol 36 (1) ◽  
pp. 69-85 ◽  
Author(s):  
Rong-Qing Jia ◽  
Charles A. Micchelli
Keyword(s):  
Finite Number ◽  
Linear Combination ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Basis Functions ◽  
Partial Difference ◽  
Integer Translates ◽  
First Case ◽  
Compactly Supported Functions ◽  
Necessary And Sufficient

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

scholarly journals The multipliers of multiple trigonometric Fourier series

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
Aizhan Ydyrys ◽  
Lyazzat Sarybekova ◽  
Nazerke Tleukhanova
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Regular System ◽  
Multiple Fourier Series ◽  
Trigonometric Fourier Series ◽  
Lorentz Spaces ◽  
Complex Numbers ◽  
Multiple Trigonometric Fourier Series

Abstract We study the multipliers of multiple Fourier series for a regular system on anisotropic Lorentz spaces. In particular, the sufficient conditions for a sequence of complex numbers {λk}k∈Zn in order to make it a multiplier of multiple trigonometric Fourier series from Lp[0; 1]n to Lq[0; 1]n , p > q. These conditions include conditions Lizorkin theorem on multipliers.

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