Abstract. We provide an orthogonal basis of polynomials for the local Dirichlet space $\mathcal {D}_\zeta $. These polynomials have numerous interesting features and a very unique algebraic pattern. We obtain the recurrence relation, the generating function, a simple formula for their norm, and explicit formulae for the distance and the orthogonal projection onto the subspace of polynomials of degree at most $n$. The latter implies a new polynomial approximation scheme in local Dirichlet spaces. Orthogonal polynomials in a harmonically weighted Dirichlet space, created by a finitely supported singular measure, are also studied.

DOI: 10.14232/actasm-021-465-4

AMS Subject Classification (1991): 30H05, 33C45, 33C47, 42B35

Keyword(s): harmonically weighted Dirichlet spaces, orthogonal polynomials, polynomial approximation

received 15.7.2021, accepted 12.9.2021. (Registered under 715/2021.)