Orthogonal projections in the local Dirichlet spaces

Abstract

We present an explicit formula for the orthogonal projection onto the subspace of analytic polynomials of degree at most $n$ in the local Dirichlet space $D_\mu$ , where the positive measure $\mu$ consists of a finite number of Dirac measures located at points on the unit circle $\mathbb T$. This result has two key aspects: first, while it is known that polynomials are dense in $D_\mu$ , this approach offers a concrete linear approximation scheme within the space. Second, due to the orthogonality of the polynomials involved, the scheme is qualitative, as the distance of an arbitrary function $f\in D_\mu$ to the projected subspace is explicitly determined.