اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAn upper bound on the Kolmogorov widths of a certain family of integral operators
89
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the family of integral operators $(K_{alpha}f)(x)$ from $L^p[0,1]$ to $L^q[0,1]$ given by $$(K_{alpha}f)(x)=int_0^1(1-xy)^{alpha -1},f(y),operatorname{d}!y, qquad 0<alpha<1.$$ The main objective is to find upper bounds for the Kolmogorov widths, where the $n$th Kolmogorov width is the infimum of the deviation of $(K_{alpha}f)$ from an $n$-dimensional subspaces of $L^p[0,1]$ (with the infimum taken over all $n$-dimensional subspaces), and is therefore a measure of how well $K_{alpha}$ can be approximated. We find upper bounds for the Kolmogorov widths in question that decrease faster than $exp(-kappa sqrt{n})$ for some positive constant $kappa$.
قيم البحث
اقرأ أيضاً
Volterra integral operators ${cal A}=sum_{k=0}^m {cal A}_k$, $({cal A}_k f)(x)= a_k (x)int_0^x t^k f(t) ,dt$, are studied acting between weighted $L_2$ spaces on $(0,+infty)$. Under certain conditions on the weights and functions $a_k$, it is shown t
hat $cal A$ is bounded if and only if each ${cal A}_k$ is bounded. This result is then applied to describe spaces of pointwise multipliers in weighted Sobolev spaces on $(0,+infty)$.
We survey recent work and announce new results concerning two singular integral operators whose kernels are holomorphic functions of the output variable, specifically the Cauchy-Leray integral and the Cauchy-SzegH o projection associated to various c
lasses of bounded domains in $mathbb C^n$ with $ngeq 2$.
The purpose of this short note is to provide a new and very short proof of a result by Sudakov, offering an important improvement of the classical result by Kolmogorov-Riesz on compact subsets of Lebesgue spaces.
Let $r(k)$ denote the maximum number of edges in a $k$-uniform intersecting family with covering number $k$. ErdH{o}s and Lovasz proved that $ lfloor k! (e-1) rfloor leq r(k) leq k^k.$ Frankl, Ota, and Tokushige improved the lower bound to $r(k) geq
left( k/2 right)^{k-1}$, and Tuza improved the upper bound to $r(k) leq (1-e^{-1}+o(1))k^k$. We establish that $ r(k) leq (1 + o(1)) k^{k-1}$.
A subset ${g_1, ldots , g_d}$ of a finite group $G$ invariably generates $G$ if the set ${g_1^{x_1}, ldots, g_d^{x_d}}$ generates $G$ for every choice of $x_i in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable
$n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The first author recently showed that $C(G)le betasqrt{|G|}$ for some absolute constant $beta$. In this paper we show that, when $G$ is soluble, then $beta$ is at most $5/3$. We also show that this is best possible. Furthermore, we show that, in general, for each $epsilon>0$ there exists a constant $c_{epsilon}$ such that $C(G)le (1+epsilon)sqrt{|G|}+c_{epsilon}$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
