Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDouble-sided Taylors approximations and their applications in Theory of analytic inequalities
65
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper the double-sided Taylors approximations are studied. A short proof of a well-known theorem on the double-sided Taylors approximations is introduced. Also, two new theorems are proved regarding the monotonicity of such approximations. Then we present some new applications of the double-sided Taylors approximations in the theory of analytic inequalities.
rate research
Read More
In this paper the double-sided Talors approximations are used to obtain generalisations and improvements of some trigonometric inequalities.
The paper summarizes the parallel session B3 {em Analytic approximations, perturbation methods, and their applications} of the GR18 conference. The talks in the session reported notably recent advances in black hole perturbations and post-Newtonian approximations as applied to sources of gravitational waves.
Let $A_infty ^+$ denote the class of one-sided Muckenhoupt weights, namely all the weights $w$ for which $mathsf M^+:L^p(w)to L^{p,infty}(w)$ for some $p>1$, where $mathsf M^+$ is the forward Hardy-Littlewood maximal operator. We show that $win A_infty ^+$ if and only if there exist numerical constants $gammain(0,1)$ and $c>0$ such that $$ w({x in mathbb{R} : , mathsf M ^+mathbf 1_E (x)>gamma})leq c w(E) $$ for all measurable sets $Esubset mathbb R$. Furthermore, letting $$ mathsf C_w ^+(alpha):= sup_{0<w(E)<+infty} frac{1}{w(E)} w({xinmathbb R:,mathsf M^+mathbf 1_E (x)>alpha}) $$ we show that for all $win A_infty ^+$ we have the asymptotic estimate $mathsf C_w ^+ (alpha)-1lesssim (1-alpha)^frac{1}{c[w]_{A_infty ^+}}$ for $alpha$ sufficiently close to $1$ and $c>0$ a numerical constant, and that this estimate is best possible. We also show that the reverse Holder inequality for one-sided Muckenhoupt weights, previously proved by Martin-Reyes and de la Torre, is sharp, thus providing a quantitative equivalent definition of $A_infty ^+$. Our methods also allow us to show that a weight $win A_infty ^+$ satisfies $win A_p ^+$ for all $p>e^{c[w]_{A_infty ^+}}$.
We address the optimal constants in the strong and the weak Stechkin inequalities, both in their discrete and continuous variants. These inequalities appear in the characterization of approximation spaces which arise from sparse approximation or have applications to interpolation theory. An elementary proof of a constant in the strong discrete Stechkin inequality given by Bennett is provided, and we improve the constants given by Levin and Stechkin and by Copson. Finally, the minimal constants in the weak discrete Stechkin inequalities and both continuous Stechkin inequalities are presented.
We study the two-weighted estimate [ bigg|sum_{k=0}^na_k(x)int_0^xt^kf(t)dt|L_{q,v}(0,infty)bigg|leq c|f|L_{p,u}(0,infty)|,tag{$*$} ] where the functions $a_k(x)$ are not assumed to be positive. It is shown that for $1<pleq qleqinfty$, provided that the weight $u$ satisfies the certain conditions, the estimate $(*)$ holds if and only if the estimate [ sum_{k=0}^nbigg|a_k(x)int_0^xt^kf(t)dt|L_{q,v}(0,infty)bigg| leq c|f|L_{p,u}(0,infty)|.tag{$**$} ] is fulfilled. The necessary and sufficient conditions for $(**)$ to be valid are well-known. The obtained result can be applied to the estimates of differential operators with variable coefficients in some weighted Sobolev spaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
