Stability Analysis of Grünwald Interpolation Operators on Chebyshev Nodes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908680565293056 |
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| author | C, Vinaya P |
| author_facet | C, Vinaya P |
| contents | In 1941, G. Grünwald proved the convergence of a sequence of operators constructed using classical Lagrange interpolation at Chebyshev nodes. In this work, we establish a perturbed version of Grünwald's result, thereby extending the class of admissible nodal points. Specifically, we provide sufficient conditions for convergence when the interpolation nodes are of the form {cos eta_k} for k = 1, ..., n, where {eta_k} is a general sequence. We refer to these operators as Grünwald operators. In particular, we prove a convergence result when {eta_k} is equidistant and uniformly distributed. We establish a Voronovskaja-type estimate for the convergence of these operators and derive quantitative results using modulus of continuity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16927 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability Analysis of Grünwald Interpolation Operators on Chebyshev Nodes C, Vinaya P Functional Analysis Classical Analysis and ODEs 41A25, 41A60, 41A17, 41A35, 41A50 In 1941, G. Grünwald proved the convergence of a sequence of operators constructed using classical Lagrange interpolation at Chebyshev nodes. In this work, we establish a perturbed version of Grünwald's result, thereby extending the class of admissible nodal points. Specifically, we provide sufficient conditions for convergence when the interpolation nodes are of the form {cos eta_k} for k = 1, ..., n, where {eta_k} is a general sequence. We refer to these operators as Grünwald operators. In particular, we prove a convergence result when {eta_k} is equidistant and uniformly distributed. We establish a Voronovskaja-type estimate for the convergence of these operators and derive quantitative results using modulus of continuity. |
| title | Stability Analysis of Grünwald Interpolation Operators on Chebyshev Nodes |
| topic | Functional Analysis Classical Analysis and ODEs 41A25, 41A60, 41A17, 41A35, 41A50 |
| url | https://arxiv.org/abs/2511.16927 |