arXiv 0905.05722009-05math.FA

Hardy和Bergman加权范数约束下的有效H^{ \infty }插值

Effective H^{\infty} interpolation constrained by Hardy and Bergman weighted norms

Rachid Zarouf

摘要：

给定单位圆盘$\mathbb{D}$的有限集$σ$和$\mathbb{D}$中属于类$X$的全纯函数$f$，我们在另一个类$Y$中寻找函数$g$，它在所有函数$g$中最小化范数$|g|_{Y}$，使得$g_{|σ}=f_{|σ}$。一般来说，所考虑的插值常数是$c(σ,\, X,\, Y)={sup}{}_{f\in X,\,\parallel f\parallel_{X}\leq1}{inf}\{|g|_{Y}:\, g_{|σ}=f_{|σ}\} \,.$。当$Y=H^{\infty}$时，我们的插值问题包括Nevanlinna-Pick（1916），Caratheodory-Schur（1908）的插值问题。此外，Carleson的自由插值（1958）也可以用我们的常数$c(σ,\, X,\, H^{\infty})$来解释。如果$X$是属于Hardy和Bergman加权空间的尺度的Hilbert空间，我们证明了$c(σ,\, X,\, H^{\infty})\leq aϕ_{X}(1-\frac{1-r}{n})$，其中$n=#σ$，$r={max}{}_{λ\inσ}|λ|$和$ϕ_{X}(t)$代表空间$X$上的评价泛函$f\mapsto f(t)$的范数.上界在给定$n$和$r$的集合$σ$上是尖锐的。如果$X$是一般的Hardy-Sobolev空间或一般的加权Bergman空间（不一定是Hilbert型的），我们也找到了$c(σ,\, X,\, H^{\infty})$（有时是特殊的集合$σ$）的上界和下界，但在这些界之间有一些差距。这种约束插值的动机是在矩阵分析和算子理论中的一些应用。

英文摘要：

Given a finite set $σ$ of the unit disc $\mathbb{D}$ and a holomorphic function $f$ in $\mathbb{D}$ which belongs to a class $X$ we are looking for a function $g$ in another class $Y$ which minimizes the norm $|g|_{Y}$ among all functions $g$ such that $g_{|σ}=f_{|σ}$. Generally speaking, the interpolation constant considered is $c(σ,\, X,\, Y)={sup}{}_{f\in X,\,\parallel f\parallel_{X}\leq1}{inf}\{|g|_{Y}:\, g_{|σ}=f_{|σ}\} \,.$ When $Y=H^{\infty}$, our interpolation problem includes those of Nevanlinna-Pick (1916), Caratheodory-Schur (1908). Moreover, Carleson's free interpolation (1958) has also an interpretation in terms of our constant $c(σ,\, X,\, H^{\infty})$.} If $X$ is a Hilbert space belonging to the scale of Hardy and Bergman weighted spaces, we show that $c(σ,\, X,\, H^{\infty})\leq aϕ_{X}(1-\frac{1-r}{n})$ where $n=#σ$, $r={max}{}_{λ\inσ}|λ|$ and where $ϕ_{X}(t)$ stands for the norm of the evaluation functional $f\mapsto f(t)$ on the space $X$. The upper bound is sharp over sets $σ$ with given $n$ and $r$.} If $X$ is a general Hardy-Sobolev space or a general weighted Bergman space (not necessarily of Hilbert type), we also found upper and lower bounds for $c(σ,\, X,\, H^{\infty})$ (sometimes for special sets $σ$) but with some gaps between these bounds.} This constrained interpolation is motivated by some applications in matrix analysis and in operator theory.}

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