Orthogonal projections in the local Dirichlet spaces
Emmanuel Fricain, Javad Mashreghi · Complex Variables and Elliptic Equations · 2025
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 Dμ, where the positive measure µ consists of a finite number of Dirac measures located at points on the unit circle T. This result has two key aspects: first, while it is known that polynomials are dense in Dμ, 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∈Dμ to the projected subspace is explicitly determined.