Uniform Convergence of Trigonometric Series with General Monotone Coefficients

2019 ◽  
Vol 71 (6) ◽  
pp. 1445-1463
Author(s):  
Mikhail Dyachenko ◽  
Askhat Mukanov ◽  
Sergey Tikhonov
Keyword(s):  
Uniform Convergence ◽  
Rate Of Convergence ◽  
Trigonometric Series ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Monotone Coefficients ◽  
Fourier Sums ◽  
Necessary And Sufficient

AbstractWe study criteria for the uniform convergence of trigonometric series with general monotone coefficients. We also obtain necessary and sufficient conditions for a given rate of convergence of partial Fourier sums of such series.

scholarly journals Norm inequalities in multidimensional Lorentz spaces

2008 ◽  
Vol 103 (2) ◽  
pp. 278
Author(s):  
Boris Simonov ◽  
Sergey Tikhonov
Keyword(s):  
Trigonometric Series ◽  
Sufficient Conditions ◽  
Lorentz Spaces ◽  
Necessary And Sufficient Conditions ◽  
Double Trigonometric Series ◽  
Norm Inequalities ◽  
Necessary And Sufficient

In this paper we obtain necessary and sufficient conditions for double trigonometric series to belong to generalized Lorentz spaces, not symmetric in general. Estimates for the norms are given in terms of coefficients.

scholarly journals On uniform convergence of some classes of infinite products

1987 ◽  
pp. 107
Author(s):  
K.M. Slepenchuk
Keyword(s):  
Uniform Convergence ◽  
Infinite Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Products ◽  
Necessary And Sufficient

We find necessary and sufficient conditions $\{ \alpha_k(x) \}$ must satisfy for the infinite product$$\prod\limits_{k=1}^{\infty} \bigl[ 1 + \alpha_k(x) u_k(x) \bigr]$$to converge uniformly under the condition that:1) the series $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)|$ converges uniformly; 2) $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)| = O(1)$.

scholarly journals On uniform convergence of infinite products

2021 ◽  
pp. 80
Author(s):  
K.M. Slepenchuk ◽  
G.A. Barbashova
Keyword(s):  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Products ◽  
Necessary And Sufficient

We establish necessary and sufficient conditions for $\{ \alpha_k(x) \}$ to satisfy such that the product $\prod\limits_{k=1}^{\infty} [1+\alpha_k(x) U_k(x)]$ converges uniformly under the condition that $\{ U_k(x) \}$ belongs to a given class.

Influence of design on the rate of convergence to normality in L1 regression

1989 ◽  
Vol 106 (2) ◽  
pp. 355-368 ◽  
Author(s):  
Peter Hall ◽  
A. H. Welsh
Keyword(s):  
Convergence Rate ◽  
Rate Of Convergence ◽  
Edgeworth Expansion ◽  
Sufficient Conditions ◽  
Error Distribution ◽  
Necessary And Sufficient Conditions ◽  
Moment Condition ◽  
Regression Problem ◽  
Design Variables ◽  
Necessary And Sufficient

AbstractWe provide a concise account of the influence of design variables on the convergence rate in an L1 regression problem. In particular, we show that the convergence rate may be characterized precisely in terms of third and fourth moments of the design variables. This result leads to necessary and sufficient conditions on the design for the Berry-Esseen rate to be achieved. We also show that a moment condition on the error distribution is necessary and sufficient for a non-uniform Berry-Esseen theorem, and that an Edgeworth expansion is possible if the design points are not too clumped.

On the Convergence of the Density of the Power Normalized Maximum1

1998 ◽  
Vol 48 (1-2) ◽  
pp. 13-20 ◽  
Author(s):  
N. R. Mohan ◽  
U. R. Subramanya
Keyword(s):  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stable Laws ◽  
Stable Law ◽  
Necessary And Sufficient ◽  
Local Uniform Convergence

Necessary and sufficient conditions for local uniform convergence of the density of the power normalized maximum to the corresponding density of a p-max stable law is derived for each of the six p-max stable laws. An unified criterian is also obtained.

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> 0 for eachj> 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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