Kernels and best approximations related to the system of ultraspherical polynomials

JOURNAL OF APPROXIMATION THEORY

卷:133

Article

Kolyada, VI ; Marcellán, F

关键词: orthogonal polynomials;    best approximations;    Nikol'skii inequalities;   

DOI  :  10.1016/j.jat.2004.12.013

来源: Elsevier

【 摘 要 】

We study the uniformly bounded orthonormal system u(lambda) of functions u(n)((lambda))(x)=phi((lambda))(n)(cos x)(sin x)(lambda), x epsilon [0,pi], where {phi((lambda))(n)}(n=0)(infinity) (lambda > 0) is the normalized system of ultraspherical polynomials. We investigate some approximation properties of the system mu(lambda) and we show that these properties are similar to one's of the trigonometric system. First, we obtain estimates of L-P-norms of the kernels of the system u(lambda). These estimates enable us to prove Nikol'skii-type inequalities for u(lambda)-polynomials. Next, we prove directly that u(lambda) is a basis in each L-w(p), 1 < p < infinity, where w is an arbitrary A(p)-weight function. Finally, we apply these results to get sharp inequalities for the best u(lambda)-approximations in L-q in terms of the best u(lambda)-approximations in L-p (1 <= p < q < infinity). For the trigonometric system such inequalities have been already known. (c) 2005 Elsevier Inc. All rights reserved.

【 授权许可】

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