Polynomial Approximation in Higher-Order Weighted Dirichlet Spaces

Abstract

This work proves an analogue of Fej\'er's Theorem for higher-order weighted Dirichlet spaces. For a holomorphic function $f \in \widehat{\mathcal{H}}_{\mu, m}$, the partial sums $S_{m,n}f$ do not generally converge unless their coefficients are strategically modified. We introduce a sequence of polynomials $p_n(z) = \sum\limits_{k=m+1}^{n+m+1} w_{n,k}a_kz^k$ with a weight array $\{w_{n,k}\}$ satisfying specific oscillation and decay properties. By leveraging the local Douglas formula and a coefficient correspondence theorem, we prove the convergence $\lim\limits_{n \to \infty} \|f - p_n\|_{\mu,m} = 0$. Notably, for a measure $\mu$ composed of Dirac masses, the adjusted coefficients admit an explicit closed-form expression.

Categories