Kernels and best approximations related to the system of ultraspherical polynomials |
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Authors: | VI Kolyada F Marcelln |
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Institution: | aDepartment of Mathematics, Karlstad University, Universitetsgatan 1, 651 88 Karlstad, Sweden;bDepartamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganés (Madrid), Spain |
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Abstract: | We study the uniformly bounded orthonormal system of functions where is the normalized system of ultraspherical polynomials. We investigate some approximation properties of the system and we show that these properties are similar to one's of the trigonometric system. First, we obtain estimates of Lp-norms of the kernels of the system . These estimates enable us to prove Nikol'skiı˘-type inequalities for -polynomials. Next, we prove directly that is a basis in each , where w is an arbitrary Ap-weight function. Finally, we apply these results to get sharp inequalities for the best -approximations in Lq in terms of the best -approximations in . For the trigonometric system such inequalities have been already known. |
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Keywords: | Orthogonal polynomials Best approximations Nikol'skiı ˘ inequalities |
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