具有多个零点和奇点的双倍权多项式逼近
Polynomial approximation with doubling weights having finitely many zeros and singularities
摘要：
我们证明了匹配的直接和逆定理（代数）多项式逼近与双重权重$w$有许多零和奇点（即，$w$变为无穷大的点），并且不会太"快速变化“远离这些零点和奇点。这类加倍权是相当广泛的，特别是，包括经典的Jacobi权，广义Jacobi权和广义Ditzian-Totik权。我们在加权$L_p$（准）范数$\|f\|_{p, w}$中用$0<p<\infty$近似，其中$\|f\|_{p, w} := \left(\int_{-1}^1 |f(u)|^p w(u) du \right)^{1/p}$。涉及相关的实现泛函的等价型结果进行了讨论。
英文摘要：
We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly changing'' away from these zeros and singularities. This class of doubling weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian-Totik weights. We approximate in the weighted $L_p$ (quasi) norm $\|f\|_{p, w}$ with $0<p<\infty$, where $\|f\|_{p, w} := \left(\int_{-1}^1 |f(u)|^p w(u) du \right)^{1/p}$. Equivalence type results involving related realization functionals are also discussed.
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