论文标题
Muckenhoupt重量产生的角林系统的Steklov问题和剩余估计
The Steklov problem and Remainder Estimates for Krein Systems generated by a Muckenhoupt weight
论文作者
论文摘要
我们表明，针对角林系统的解决方案，单位圆上正交多项式的连续频率类似物，由$ a_2（\ m athbb {r}）$ w $ w $满足$ w-1 \ in L^1（\ Mathbb {r}） + l^2（\ Mathbb {\ Mathbb {\ riparty riparty， $ l^p _ {\ mathrm {loc}}}（w，\ mathbb {r}）$ for $ p $足够接近$ 2 $。这为凯林系统的Steklov问题提供了积极的答案。此外，我们定义了一个“剩余”，该“剩余”可以测量孔素系统的解决方案与多项式近似近似的差异，并且我们在$ l^p_w中估计这些剩余物（\ sathbb {r}）$ for $ w \ for $ w \ for a_2（\ mathbb {r}）中的$ w \ for $ w \ for $ w \ for $ w \ for $ w \这种多项式样近似值及其剩余的估计似乎是孔林系统独有的，在单位圆上没有类似于正交多项式的类似物。
We show that solutions to Krein systems, the continuous frequency analogue of orthogonal polynomials on the unit circle, generated by an $A_2 (\mathbb{R})$ weight $w$ satisfying $w-1 \in L^1 (\mathbb{R}) + L^2 (\mathbb{R})$, are uniformly bounded in $L^p_{\mathrm{loc}} (w, \mathbb{R})$ for $p$ sufficiently close to $2$. This provides a positive answer to the Steklov problem for Krein systems. Furthermore, we define a "remainder" which measures the difference between the solution to a Krein system and a polynomial-like approximant, and we estimate these remainders in $L^p_w (\mathbb{R})$ for $w \in A_2 (\mathbb{R})$ satisfying some additional conditions. Such polynomial-like approximants, and hence remainder estimates, seem unique to Krein systems, with no analogue for orthogonal polynomials on the unit circle.
