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 Google™ Scholar. Others By: Kolyada, V. I. - Marcellán, Francisco
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 Title: Kernels and best approximations related to the system of ultraspherical polynomials Author(s): Kolyada, V. I.Marcellán, Francisco Publisher: Elsevier Issued date: Apr-2005 Citation: Journal of Approximation Theory, 2005, vol. 133, n. 2, p. 173-194 URI: http://hdl.handle.net/10016/5952 ISSN: 0021-9045 DOI: 10.1016/j.jat.2004.12.013 Description: 22 pages, no figures.-- MSC2000 codes: 41A10; 41A17; 41A25; 42C10; 33C45.MR#: MR2129477 (2005k:41087)Zbl#: Zbl 1074.41003 Abstract: We study the uniformly bounded orthonormal system $\Cal U_\lambda$ of functions $$u_n (\lambda )}(x)=\varphi _n (\lambda )}(\cos x) (\sin x) lambda , \quad x \in [0,\pi],$$ where $\{\varphi _n (\lambda )} \}_{n=0} infty$ ($\lambda \geq 0$) is the normalized system of ultraspherical polynomials. We investigate some approximation properties of the system $\Cal U_\lambda$ and we show that these properties are similar to one's of the trigonometric system. First, we obtain estimates of $L$-norms of the kernels of the system $\Cal U_\lambda$. These estimates enable us to prove Nikol'skij-type inequalities for $\Cal U_\lambda$-polynomials. Next, we prove directly that $\Cal U_\lambda$ is a basis in each $L _w$, \$1

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