Orthogonal projections in the local Dirichlet spaces

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 $D_μ$ , where the positive measure $μ$ 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 $D_μ$ , 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 $f\in D_μ$ to the projected subspace is explicitly determined.

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