Approximation by Modified Taylor Polynomials in the Harmonically Weighted Dirichlet Spaces

In a harmonically weighted Dirichlet space \({\mathcal {D}}_\mu \) ${\mathcal{D}}_{\mu }$ , where \(\mu \) $\mu$ is a finitely supported discrete measure, Taylor polynomials of a function \(f \in {\mathcal {D}}_\mu \) $f\in {\mathcal{D}}_{\mu }$ do not necessarily converge. We show that by properly modifying a fix number of final coefficients in Taylor polynomials of f, a new sequence is created which converges to f. The number of coefficients to be modified is at most equal to the cardinality of the support of \(\mu \) $\mu$ , a fact which was first observed in (Mashreghi and Ransford in Complex Anal Synerg 7:1-11, 2021) for the special case of cardinality one. We also show that the nth modified polynomial can be interpreted as the orthogonal projection onto the subspace of polynomials of degree n under a suitable norm on \({\mathcal {D}}_\mu \) ${\mathcal{D}}_{\mu }$ .