Exact $L_2$ Bernstein-Markov inequalities for generalized weights
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909403875115008 |
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| author | Li, Jiansong Geng, Jiaxin Ling, Yun Wang, Heping |
| author_facet | Li, Jiansong Geng, Jiaxin Ling, Yun Wang, Heping |
| contents | In this paper, we obtain some exact $L_2$ Bernstein-Markov inequalities for generalized Hermite and Gegenbauer weight. More precisely, we determine the exact values of the extremal problem $$M_n^2(L_2(W_λ),{\rm D}):=\sup_{0\neq p\in\mathcal{P}_n}\frac{\int_I\left|{\rm D} p(x)\right|^2W_λ(x){\rm d}x}{\int_I| p(x)|^2W_λ(x){\rm d}x},\ λ>0,$$ where $\mathcal{P}_n$ denotes the set of all algebraic polynomials of degree at most $n$, ${\rm D}$ is the differential operator given by $${\rm D}=\Bigg\{\begin{aligned}&\frac {\rm d}{{\rm d}x}\ {\rm or}\ \mathcal{D}_λ, &&{\rm if}\ W_λ(x)=|x|^{2λ}e^{-x^2}\ {\rm and}\ I=\mathbb R, \\&(1-x^2)^{\frac12}\,\frac {\rm d}{{\rm d}x}\ {\rm or}\ (1-x^2)^{\frac12}\,\mathcal{D}_λ, &&{\rm if}\ W_λ(x):=|x|^{2λ}(1-x^2)^{μ-\frac 12},μ>-\frac12\ {\rm and}\ I=[-1,1],\end{aligned} $$ and $\mathcal{D}_λ$ is the univariate Dunkl operator, i.e., $\mathcal{D}_λf(x)=f'(x)+λ{(f(x)-f(-x))}/{x}$. Furthermore, the corresponding extremal polynomials are also obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16359 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact $L_2$ Bernstein-Markov inequalities for generalized weights Li, Jiansong Geng, Jiaxin Ling, Yun Wang, Heping Classical Analysis and ODEs 33C45, 41A17, 41A44, 42C05 In this paper, we obtain some exact $L_2$ Bernstein-Markov inequalities for generalized Hermite and Gegenbauer weight. More precisely, we determine the exact values of the extremal problem $$M_n^2(L_2(W_λ),{\rm D}):=\sup_{0\neq p\in\mathcal{P}_n}\frac{\int_I\left|{\rm D} p(x)\right|^2W_λ(x){\rm d}x}{\int_I| p(x)|^2W_λ(x){\rm d}x},\ λ>0,$$ where $\mathcal{P}_n$ denotes the set of all algebraic polynomials of degree at most $n$, ${\rm D}$ is the differential operator given by $${\rm D}=\Bigg\{\begin{aligned}&\frac {\rm d}{{\rm d}x}\ {\rm or}\ \mathcal{D}_λ, &&{\rm if}\ W_λ(x)=|x|^{2λ}e^{-x^2}\ {\rm and}\ I=\mathbb R, \\&(1-x^2)^{\frac12}\,\frac {\rm d}{{\rm d}x}\ {\rm or}\ (1-x^2)^{\frac12}\,\mathcal{D}_λ, &&{\rm if}\ W_λ(x):=|x|^{2λ}(1-x^2)^{μ-\frac 12},μ>-\frac12\ {\rm and}\ I=[-1,1],\end{aligned} $$ and $\mathcal{D}_λ$ is the univariate Dunkl operator, i.e., $\mathcal{D}_λf(x)=f'(x)+λ{(f(x)-f(-x))}/{x}$. Furthermore, the corresponding extremal polynomials are also obtained. |
| title | Exact $L_2$ Bernstein-Markov inequalities for generalized weights |
| topic | Classical Analysis and ODEs 33C45, 41A17, 41A44, 42C05 |
| url | https://arxiv.org/abs/2411.16359 |