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Polynomial Approximation in Higher-Order Weighted Dirichlet Spaces

Fejér's theorem guarantees norm convergence of Cesàro means of Taylor partial sums in the Hardy space, whereas such convergence generally fails in weighted Dirichlet-type spaces, especially in the higher-order setting. In this paper, we investigate summability problems in higher-order weighted Dirichlet spaces $\widehat{\mathcal{H}}_{μ,m}$ and show that Taylor partial sums are not uniformly bounded in these spaces and may therefore diverge in norm. To restore convergence, we introduce a family of modified polynomials whose coefficients are adjusted by a suitable weight array. Under mild boundedness and variation assumptions on the weights, we establish norm convergence of the modified sums via a coefficient correspondence principle and a Local Douglas formula. As an application, when the weight measure $μ$ is a finite sum of Dirac point masses, explicit formulas for the modified coefficients are obtained, yielding a Fejér-type summability theorem for higher-order weighted Dirichlet spaces.