Orthogonal projections in the local Dirichlet spaces

TL;DR

This paper derives an explicit formula for orthogonal projections onto polynomial subspaces in local Dirichlet spaces with finite Dirac measures, enabling concrete approximation schemes and explicit distance calculations.

Contribution

It provides a new explicit formula for orthogonal projections in local Dirichlet spaces with finite Dirac measures, facilitating concrete approximation and analysis.

Findings

Explicit projection formula for polynomial subspaces in $D_rac{$ spaces.

Concrete linear approximation scheme within the space.

Explicit calculation of the distance to the projected subspace.

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.