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 $h^{\omega}(\mathbb{R})$.

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 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 $\alpha$ ($0 < \alpha < 1$).

scholarly journals Admissibility of Fractional Order Descriptor Systems Based on Complex Variables: An LMI Approach

2020 ◽  
Vol 4 (1) ◽  
pp. 8
Author(s):  
Xuefeng Zhang ◽  
Yuqing Yan
Keyword(s):  
Fractional Order ◽  
Complex Plane ◽  
Sufficient Conditions ◽  
Complex Variables ◽  
Necessary And Sufficient Conditions ◽  
Descriptor Systems ◽  
Fractional Order Systems ◽  
Lmi Approach ◽  
Numerical Examples ◽  
Necessary And Sufficient

This paper is devoted to the admissibility issue of singular fractional order systems with order α ∈ ( 0 , 1 ) based on complex variables. Firstly, with regard to admissibility, necessary and sufficient conditions are obtained by strict LMI in complex plane. Then, an observer-based controller is designed to ensure system admissible. Finally, numerical examples are given to reveal the validity of the theoretical conclusions.

scholarly journals The Oscillation of a Class of the Fractional-Order Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qianli Lu ◽  
Feng Cen
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Analysis Method ◽  
Constant Coefficients ◽  
Delay Differential ◽  
Necessary And Sufficient

Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this,α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.

Overconvergence of series in generalized mittag-leffler functions

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Integral Equations ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Differential And Integral Equations

AbstractSeries defined by means of the three-parametric Mittag-Leffler functions, called also the Prabhakar functions, are considered in this paper. Their behaviour is investigated on the boundaries of the convergence domains. Necessary and sufficient conditions for their overconvergence are proposed. The corresponding results for series in Mittag-Leffler functions are discussed as a particular case. Such kind of results are motivated by the fact that the solutions of some fractional order differential and integral equations can be written in terms of series (or series of integrals) of Mittag-Leffler type functions.

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