Approximation by Modified Taylor Polynomials in the Harmonically Weighted Dirichlet Spaces

In a harmonically weighted Dirichlet space D mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}_\mu $$\end{document}, where mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is a finitely supported discrete measure, Taylor polynomials of a function f is an element of D mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in {\mathcal {D}}_\mu $$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}, 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 D mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}_\mu $$\end{document}.