Kernels and best approximations related to the system of ultraspherical polynomials

We study the uniformly bounded orthonormal system Uλ of functions un(λ)(x)=ϕn(λ)(cosx)(sinx)λ,x∈[0,π],where {ϕn(λ)}n=0∞(λ>0) is the normalized system of ultraspherical polynomials. We investigate some approximation properties of the system Uλ 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 Uλ. These estimates enable us to prove Nikol'skiı˘-type inequalities for Uλ-polynomials. Next, we prove directly that Uλ is a basis in each Lwp,1