We prove that for entire functions f of finite order, there is a sequence of integers \(\mathcal {S}\) such that as \(n\rightarrow \infty \) through S,

$$\begin{aligned} \min \left\{ \left| f-\left[ n/n\right] \right| \left( z\right) ,\left| f-\left[ n-1/n-1\right] \right| \left( z\right) \right\} ^{1/n}\rightarrow 0 \end{aligned}$$

uniformly for z in compact subsets of the plane. More generally this holds for sequences of Newton–Padé approximants and for functions whose errors of approximation by rational functions of type \(\left( n,n\right) \) decay sufficiently fast. This establishes George Baker’s patchwork conjecture for large classes of entire functions.

中文翻译：

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关于对角Padé近似的Baker的拼凑猜想

我们证明，对于有限阶的整个函数f，存在一个整数\（\ mathcal {S} \）序列，例如\（n \ rightarrow \ infty \）到S，

$$ \ begin {aligned} \ min \ left \ {\ left | f- \ left [n / n \ right] \ right | \ left（z \ right），\ left | f- \ left [n-1 / n-1 \ right] \ right | \ left（z \ right）\ right \} ^ {1 / n} \ rightarrow 0 \ end {aligned} $$

在平面的紧凑子集中对z均匀。更一般而言，这适用于牛顿-帕德近似值的序列以及因\（\ left（n，n \ right）\）类型的有理函数引起的近似误差衰减足够快的函数。这建立了乔治·贝克对大类整体功能的拼凑猜想。