Kernels and best approximations related to the system of ultraspherical polynomials

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 $L _w$, $1<p<\infty$, where $w$ is an arbitrary $A_p$-weight function. Finally, we apply these results to get sharp inequalities for the best $\Cal U_\lambda $-approximations in $L $ in terms of the best $\Cal U_\lambda $-approximations in $L $ ($1\leq p<q<\infty$). For the trigonometric system such inequalities have been already known.