F-多项式Pade插值与超限直径
Pade interpolation by F-polynomials and transfinite diameter
摘要：
我们定义$F$ -多项式为整函数$F$的某些频率的伸缩的线性组合。本文利用单位圆盘上全纯函数的Pade插值，利用$F$ -多项式得到了显式逼近的$F$ -多项式，并给出了其系数的精确估计.我们表明，当频率位于一个紧凑的集合$K\subset\mathbb C$，然后最佳的选择插值多项式的频率是类似的Fekete点。此外，插值算子的最小范数形成一个序列，其增长率由$K$的超限直径决定。对于$K$上测度的拉普拉斯变换，证明了在频率为Fekete点的条件下，插值多项式的系数保持有界.最后，给出了单位圆上测度的一个充分条件，它保证插值多项式系数的绝对值和有界。
英文摘要：
We define $F$-polynomials as linear combinations of dilations by some frequencies of an entire function $F$. In this paper we use Pade interpolation of holomorphic functions in the unit disk by $F$-polynomials to obtain explicitly approximating $F$-polynomials with sharp estimates on their coefficients. We show that when frequencies lie in a compact set $K\subset\mathbb C$ then optimal choices for the frequencies of interpolating polynomials are similar to Fekete points. Moreover, the minimal norms of the interpolating operators form a sequence whose rate of growth is determined by the transfinite diameter of $K$. In case of the Laplace transforms of measures on $K$, we show that the coefficients of interpolating polynomials stay bounded provided that the frequencies are Fekete points. Finally, we give a sufficient condition for measures on the unit circle which ensures that the sums of the absolute values of the coefficients of interpolating polynomials stay bounded.
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