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.

scholarly journals On convergence of Fourier integrals and Lipschitz spaces defined with differences of fractional order

2015 ◽  
Vol 23 ◽  
pp. 75
Author(s):  
B.I. Peleshenko ◽  
T.N. Semirenko
Keyword(s):  
Fractional Order ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lipschitz Spaces ◽  
Lipschitz Classes ◽  
Necessary And Sufficient ◽  
Fourier Integrals

The necessary and sufficient conditions in terms of Fourier transforms $\hat{f}$ of functions $f\in L^1(\mathbb{R})$ are obtained for $f$ to belong to the Lipschitz classes $H^{\omega}(\mathbb{R})$, $h^{\omega}(\mathbb{R})$.

scholarly journals Absolute convergence of Fourier integrals and Lipschitz classes defined with differences of fractional order

2013 ◽  
Vol 21 ◽  
pp. 145
Author(s):  
B.I. Peleshenko ◽  
T.N. Semirenko
Keyword(s):  
Fractional Order ◽  
Absolute Convergence ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lipschitz Classes ◽  
Necessary And Sufficient ◽  
Fourier Integrals

The necessary and sufficient conditions in terms of Fourier transforms $\hat{f}$ of functions $f\in L^1(\mathbb{R})$ are obtained for $f$ to belong to the Lipschitz classes $H_C^{\omega, \alpha}(\mathbb{R})$ and $h_C^{\omega, \alpha}(\mathbb{R})$, defined by differences of fractional order.

scholarly journals Absolute convergence of Fourier integrals and Lipschitz classes

2021 ◽  
Vol 19 ◽  
pp. 102
Author(s):  
B.I. Peleshenko
Keyword(s):  
Absolute Convergence ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lipschitz Classes ◽  
Necessary And Sufficient ◽  
Fourier Integrals

The necessary and sufficient conditions, in terms of Fourier transforms $\hat{f}$ of functions $f \in L^1(\mathbb{R})$, are obtained for $f$ to belong to the Lipschitz classes $H^{\omega}(\mathbb{R})$ and $h^{\omega}(\mathbb{R})$.

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 Lipschitz conditions and lacunarity

1973 ◽  
Vol 16 (3) ◽  
pp. 272-277
Author(s):  
R. E. Edwards
Keyword(s):  
Fourier Series ◽  
Fourier Coefficients ◽  
Lacunary Series ◽  
Periodic Functions ◽  
Lipschitz Classes ◽  
Special Case ◽  
Lipschitz Conditions

We consider 2π-periodic functions on the limits and give simple and complete characterizations, in terms of Fourier coefficients, of functions which belong to various Lipschitz classes and whose Fourier series are lacunary. Such characterisations seem to be missing from the literature, though there are various wellknown partial characterisations valid for functions with arbitrary spectra; cf. the remarks following Theorem 1. The results given below form complements to and sharpenings of some of the standard results valid for the special case of lacunary series.

scholarly journals Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Fractional Derivative ◽  
Sufficient Conditions ◽  
Variational Problems ◽  
Necessary And Sufficient Conditions ◽  
Optimal Conditions ◽  
Lagrange Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Necessary And Sufficient

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.

Semidirect Product Compactifications

1983 ◽  
Vol 35 (1) ◽  
pp. 1-32
Author(s):  
F. Dangello ◽  
R. Lindahl
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Topological Semigroup ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Semigroups ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

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