The Cesàro means of Taylor polynomials \(\sigma _n,\) \(n \ge 0,\) are finite rank operators on any Banach space of analytic functions on the open unit disc. They are particularly exploited when the Taylor polynomials do not constitute a valid linear polynomial approximation scheme (LPAS). Notably, in local Dirichlet spaces \({\mathcal {D}}_\zeta ,\) they serve as a proper LPAS. The primary objective of this note is to accurately determine the norm of \(\sigma _n\) when it is considered as an operator on \({\mathcal {D}}_\zeta .\) There exist several practical methods to impose a norm on \({\mathcal {D}}_\zeta ,\) and each norm results in a distinct operator norm for \(\sigma _n.\) In this context, we explore three different norms on \({\mathcal {D}}_\zeta \) and, for each norm, precisely compute the value of \(\Vert \sigma _n\Vert _{{\mathcal {D}}_\zeta \rightarrow {\mathcal {D}}_\zeta }.\) Furthermore, in all instances, we identify the maximizing functions and demonstrate their uniqueness.

泰勒多项式\(\sigma _n,\) \(n \ge 0,\)的 Cesàro 均值是开单位圆盘上解析函数的任何 Banach 空间上的有限秩算子。当泰勒多项式不构成有效的线性多项式逼近方案（LPAS）时，它们尤其被利用。值得注意的是，在局部狄利克雷空间\({\mathcal {D}}_\zeta ,\)中，它们充当适当的 LPAS。本文的主要目的是准确确定\(\sigma _n\)的范数，当它被视为\({\mathcal {D}}_\zeta 上的运算符时。\)存在几种实用的方法来强加\({\mathcal {D}}_\zeta ,\)上的范数，每个范数都会导致\(\sigma _n.\)的不同算子范数。在这种情况下，我们探索\({\ mathcal {D}}_\zeta \)并针对每个范数，精确计算\(\Vert \sigma _n\Vert _{{\mathcal {D}}_\zeta \rightarrow {\mathcal {D} }_\zeta }.\)此外，在所有情况下，我们都识别了最大化函数并证明了它们的独特性。