Best approximation by polynomials on the conic domains
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913917213605888 |
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| author | Ge, Yan Xu, Yuan |
| author_facet | Ge, Yan Xu, Yuan |
| contents | A new modulus of smoothness and its equivalent $K$-function are defined on the conic domains in $\mathbb{R}^d$, and used to characterize the weighted best approximation by polynomials. Both direct and weak inverse theorems of the characterization are established via the modulus of smoothness. For the conic surface $\mathbb{V}_0^{d+1} = \{(x,t): \|x\| = t\le 1\}$, the natural weight function is $t^{-1}(1-t)^γ$, which has a singularity at the apex, the rotational part of the modulus of smoothness is defined in terms of the difference operator in Euler angles with an increment $h/\sqrt{t}$, akin to the Ditzian-Totik modulus on the interval but with $\sqrt{t}$ in the denominator, which captures the singularity at the apex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22916 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Best approximation by polynomials on the conic domains Ge, Yan Xu, Yuan Classical Analysis and ODEs 41A10, 41A63, 42C10, 42C40 A new modulus of smoothness and its equivalent $K$-function are defined on the conic domains in $\mathbb{R}^d$, and used to characterize the weighted best approximation by polynomials. Both direct and weak inverse theorems of the characterization are established via the modulus of smoothness. For the conic surface $\mathbb{V}_0^{d+1} = \{(x,t): \|x\| = t\le 1\}$, the natural weight function is $t^{-1}(1-t)^γ$, which has a singularity at the apex, the rotational part of the modulus of smoothness is defined in terms of the difference operator in Euler angles with an increment $h/\sqrt{t}$, akin to the Ditzian-Totik modulus on the interval but with $\sqrt{t}$ in the denominator, which captures the singularity at the apex. |
| title | Best approximation by polynomials on the conic domains |
| topic | Classical Analysis and ODEs 41A10, 41A63, 42C10, 42C40 |
| url | https://arxiv.org/abs/2506.22916 |