Abstract
The partial Taylor sums \(S_n\), \(n \ge 0\), are finite rank operators on any Banach space of analytic functions on the open unit disc. In the classical setting of disc algebra \({\mathcal {A}}\), the precise value of \(\Vert S_n\Vert _{{\mathcal {A}} \rightarrow {\mathcal {A}}}\) is not known. These numbers are referred as the Lebesgue constants and they grow like \(\log n\), modulo a multiplicative constant, when n tends to infinity. In this note, we study \(\Vert S_n\Vert \) when it acts on the local Dirichlet space \({\mathcal {D}}_\zeta \). There are several distinguished ways to put a norm on \({\mathcal {D}}_\zeta \) and each choice naturally leads to a different operator norm for \(S_n\), as an operator on \({\mathcal {D}}_\zeta \). We consider three different norms on \({\mathcal {D}}_\zeta \) and, in each case, evaluate the precise value of \(\Vert S_n\Vert _{{\mathcal {D}}_\zeta \rightarrow {\mathcal {D}}_\zeta }\). In all cases, we also show that the maximizing function is unique. These formulas indicate that \(\Vert S_n\Vert _{{\mathcal {D}}_\zeta \rightarrow {\mathcal {D}}_\zeta } \asymp \sqrt{n}\) as n grows. Hence, in the light of uniform boundedness principle, there is a function \(f \in {\mathcal {D}}_\zeta \) such that the local sequence \(\Vert S_nf\Vert _{{\mathcal {D}}_{\zeta }}\), \(n \ge 1\), is unbounded. We provide two explicit constructions.
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Mashreghi, J., Shirazi, M. & Withanachchi, M. Partial Taylor sums on local Dirichlet spaces. Anal.Math.Phys. 12, 147 (2022). https://doi.org/10.1007/s13324-022-00756-9
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DOI: https://doi.org/10.1007/s13324-022-00756-9