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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.26133 |
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| _version_ | 1866917174762799104 |
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| author | Yan, Yuanhao He, Li |
| author_facet | Yan, Yuanhao He, Li |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26133 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial Approximation in Higher-Order Weighted Dirichlet Spaces Yan, Yuanhao He, Li Functional Analysis 32E30, 32A37 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. |
| title | Polynomial Approximation in Higher-Order Weighted Dirichlet Spaces |
| topic | Functional Analysis 32E30, 32A37 |
| url | https://arxiv.org/abs/2510.26133 |