Geodesic
Bounded point derivations on certain function spaces
Algebra i analiz, Tome 31 (2019) no. 2, pp. 174-188.

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Let $ X$ be a compact nowhere dense subset of the complex plane $ \mathbb{C}$, and let $ dA$ denote two-dimensional Lebesgue (or area) measure in $ \mathbb{C}$. Denote by $ \mathcal {R}(X)$ the set of all rational functions having no poles on $ X$, and by $ R^p(X)$ the closure of $ \mathcal {R}(X)$ in $ L^p(X,dA)$ whenever $ 1\leq p\infty $. The purpose of this paper is to study the relationship between bounded derivations on $ R^p(X)$ and the existence of approximate derivatives provided $ 2$, and to draw attention to an anomaly that occurs when $ p=2$.
Keywords: point derivation, approximate derivative, monogeneity, capacity. 
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J. E. Brennan. Bounded point derivations on certain function spaces. Algebra i analiz, Tome 31 (2019) no. 2, pp. 174-188. http://geodesic.mathdoc.fr/item/AA_2019_31_2_a6/

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