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VIDEO LIBRARY |
International conference "High-dimensional approximation and discretization"
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Nikolskii constants for spherical polynomials and entire functions of spherical exponential type D. V. Gorbachev |
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Abstract: We study the asymptotic behavior of sharp Nikolskii constant $$\mathcal{C}(n, d, p, q) := \sup \{\|{f}\|_{L^q(\mathbb{S}^d)}: f \in \Pi_n^d, \|{f}\|_{L^p(\mathbb{S}^d)} = 1\}$$ for 1. We prove that for $$\lim\limits_{n \to \infty} \frac{\mathcal{C}(n, d, p, \infty)} {n^{d/p}} = \mathcal{L}(d, p, \infty),$$ and for $$\liminf\limits_{n \to \infty} \frac{\mathcal{C}(n, d, p, q)} {n^{d (1/p-1/q)}} \ge \mathcal{L}(d, p, q),$$ where the constant $$\mathcal{L}(d, p, q) := \sup \{\|{f}\|_{L^q(\mathbb{R}^d)}: f \in \mathcal{E}_p^d, \|{f}\|_{L^p(\mathbb{R}^d)} = 1\}$$ with These results extend the recent results of Levin and Lubinsky for trigonometric polynomials on the unit circle. Compared with those in one variable, our proof in higher-dimensional case is more difficult because functions on the sphere can not be identified as periodic functions on Euclidean space and explicit connections between spherical polynomial interpolation and the Shannon sampling theorem for entire functions of exponential type are not available. Our proof of the upper estimate relies on a recent deep result of Bondarenko, Radchenko and Viazovska on spherical designs: $$\frac{1}{|\mathbb{S}^d|} \int\limits_{\mathbb{S}^d} f(x) dx = \frac{1}{N} \sum\limits^{N}_{j=1} f(x_{n,j} ),\quad f \in \Pi_n^d,$$ an earlier result of Yudin on the distribution of points of spherical designs $$\|{f}\|_p \asymp \left(\sum\limits_{\omega\in\Lambda} \lambda_{n,j} |f(x_{n,j} )|^p \right)^{1/p} ,\quad 0 < p < \infty.$$ The proof of the lower estimate is based on the de la Vallée-Poussin type kernels associated with a smooth cutoff function on the sphere and also some properties of the exponential mapping from the tangent plane to the sphere, which connects functions on sphere with functions on Euclidean space. 2. While it remains a very challenging open problem to determine the exact value of the Nikolskii constant $$\sup\limits_{0\le f\in\mathcal{E}^d_1} \frac{\|{f}\|_{L^{\infty}(\mathbb{R}^d)}}{\|{f}\|_{L^{1}(\mathbb{R}^d)}} = \frac{1}{2^{d-1}|\mathbb{S}^d|\Gamma(d + 1)}.$$ 3. We investigate the normalized Nikolskii constant $$L_d :=\frac{|\mathbb{S}^d|\Gamma(d + 1)}{2}\mathcal{L}(d, 1,\infty).$$ For this problem, we first show existence of an extremal function and its uniqueness. It was known that $$2^{-d} \le L_d \le {}_1F_2\left( \frac{d}{2};\frac{d}{2}+1;\frac{d}{2}+1;-\frac{\beta^2_d}{4}\right),$$ where $$(0.5)^d \le L_d \le (\sqrt{2/e})^{d(1+o(1))},\quad \sqrt{2/e} = 0.85776 \dots .$$ 4. We observe that for Language: English |