scholarly journals A note on the norm convergence by Vilenkin–Fejér means

2014 ◽  
Vol 21 (4) ◽  
Author(s):  
George Tephnadze
Keyword(s):  
Modulus Of Continuity ◽  
Hardy Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Norm Convergence ◽  
Fejér Means ◽  
Necessary And Sufficient

AbstractThe main aim of this paper is to find necessary and sufficient conditions for the convergence of Fejér means in terms of the modulus of continuity on the Hardy spaces

Modulus of continuity and convergence of subsequences of Vilenkin–Fejér means in martingale Hardy spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Giorgi Tutberidze
Keyword(s):  
Hardy Space ◽  
Modulus Of Continuity ◽  
Hardy Spaces ◽  
Lebesgue Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Fejér Means ◽  
Necessary And Sufficient

Abstract In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 < p < 1 2 {0<p<\frac{1}{2}} .

Uniform and 𝐿-Convergence of Logarithmic Means of Double Walsh–Fourier Series

2005 ◽  
Vol 12 (1) ◽  
pp. 75-88
Author(s):  
György Gát ◽  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Function Space ◽  
Modulus Of Continuity ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Two Dimensional ◽  
Logarithmic Means ◽  
Convergence And Divergence ◽  
Necessary And Sufficient ◽  
Series Of Functions

Abstract We discuss some convergence and divergence properties of twodimensional (Nörlund) logarithmic means of two-dimensional Walsh–Fourier series of functions both in the uniform and in the Lebesgue norm. We give necessary and sufficient conditions for the convergence regarding the modulus of continuity of the function, and also the function space.

scholarly journals Toeplitz operators and Wiener-Hopf factorisation: an introduction

2017 ◽  
Vol 4 (1) ◽  
pp. 130-145 ◽  
Author(s):  
M. Cristina Câmara
Keyword(s):  
Integral Equations ◽  
Half Plane ◽  
Singular Integral ◽  
Hardy Spaces ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Necessary And Sufficient Conditions ◽  
Upper Half Plane ◽  
Necessary And Sufficient

Abstract Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces Hp of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in L−1(ℝ)

scholarly journals On the Modulus of Continuity of Mappings Between Euclidean Spaces

2013 ◽  
Vol 112 (1) ◽  
pp. 147
Author(s):  
Dieudonné Agbor ◽  
Jan Boman
Keyword(s):  
Modulus Of Continuity ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Euclidean Spaces ◽  
The Real ◽  
Lipschitz Continuous ◽  
Finite Set ◽  
Necessary And Sufficient

Let $f$ be a function from $\mathbf{R}^p$ to $\mathbf{R}^q$ and let $\Lambda$ be a finite set of pairs $(\theta, \eta) \in \mathbf{R}^p \times \mathbf{R}^q$. Assume that the real-valued function $\langle\eta, f(x)\rangle$ is Lipschitz continuous in the direction $\theta$ for every $(\theta, \eta) \in \Lambda$. Necessary and sufficient conditions on $\Lambda$ are given for this assumption to imply each of the following: (1) that $f$ is Lipschitz continuous, and (2) that $f$ is continuous with modulus of continuity $\le C\epsilon |{\log \epsilon}|$.

Norm Convergence of Tn

1977 ◽  
Vol 29 (6) ◽  
pp. 1340-1344 ◽  
Author(s):  
Glenn R. Luecke
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Necessary And Sufficient Conditions ◽  
Norm Convergence ◽  
Continuous Linear ◽  
Linear Transformations ◽  
Necessary And Sufficient

Throughout this paper X will denote a complex Banach space and all operators T will be assumed to be continuous linear transformations from X into X. If T is an operator then ┘(T), γ(T), and R(T) will denote the spectrum of T, the spectral radius of T, and range of T, respectively. This paper contains necessary and sufficient conditions for the (norm) convergence of {Tn} when T is an operator on X.

scholarly journals Composition Operators Mapping Logarithmic Bloch Functions into Hardy Space

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Sufficient Conditions ◽  
Hyperbolic Type ◽  
Necessary And Sufficient Conditions ◽  
Bloch Spaces ◽  
Bloch Functions ◽  
Hardy Classes ◽  
Necessary And Sufficient

Characterizing the hyperbolic Hardy classes, several g-functions of hyperbolic type are introduced. Using this, necessary and sufficient conditions on the inducing self-maps are established for the boundedness of the composition operators from logarithmic Bloch spaces into Hardy spaces.

scholarly journals Generalized Hausdorff Operators on K ̇ α , q β , p ℝ and H K ̇ α , q β , p , N ℝ in the Dunkl Settings

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Faouaz Saadi ◽  
Othman Tyr ◽  
Radouan Daher
Keyword(s):  
Hardy Spaces ◽  
Hardy Inequality ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cesàro Operator ◽  
Herz Spaces ◽  
Hausdorff Operators ◽  
Necessary And Sufficient ◽  
Cesaro Operator

In the present paper, we obtain some new results, and we generalize some known results for the Hausdorff operators. We have studied the generalized Hausdorff operators H α , φ on the Dunkl-type homogeneous weighted Herz spaces K ̇ α , q β , p ℝ and Dunkl Herz-type Hardy spaces H K ̇ α , q β , p , N ℝ . We have determined simple sufficient conditions for these operators to be bounded on these spaces. As applications, we provide necessary and sufficient conditions for generalized Cesàro operator to be bounded on K ̇ α , q β , p ℝ and Hardy inequality for K ̇ α , q β , p ℝ .

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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