Limit Ultraspherical Series and Their Approximation Properties

We study new series of the form $\sum_{k=0}^\infty f_k^{-1} \widehat P_k^{-1}(x)$ in which the general term $f_k^{-1}\widehat P_k^{-1}(x)$, $k=0,1,\dots$, is obtained by passing to the limit as $\alpha\to-1$ from the general term $\widehat f_k^\alpha\widehat P_k^{\alpha,\alpha}(x)$ of the Fourier series $\sum_{k=0}^\infty f_k^\alpha\widehat P_k^{\alpha,\alpha}(x)$ in Jacobi ultraspherical polynomials $\widehat P_k^{\alpha,\alpha}(x)$ generating, for $\alpha>-1$, an orthonormal system with weight $(1-x^2)^\alpha$ on $[-1,1]$. We study the properties of the partial sums $S_n^{-1}(f,x)=\sum_{k=0}^nf_k^{-1}\widehat P_k^{-1}(x)$ of the limit ultraspherical series $\sum_{k=0}^\infty f_k^{-1}\widehat P_k^{-1}(x)$. In particular, it is shown that the operator $S_n^{-1}(f)=S_n^{-1}(f,x)$ is the projection onto the subspace of algebraic polynomials $p_n=p_n(x)$ of degree at most $n$, i.e., $S_n(p_n)=p_n$; in addition, $S_n^{-1}(f,x)$ coincides with $f(x)$ at the endpoints $\pm1$, i.e., $S_n^{-1}(f,\pm1)=f(\pm1)$. It is proved that the Lebesgue function $\Lambda_n(x)$ of the partial sums $S_n^{-1}(f,x)$ is of the order of growth equal to $O(\ln n)$, and, more precisely, it is proved that $\Lambda_n(x)\le c(1+\ln(1+n\sqrt{1-x^2}\mspace{2mu}))$, $-1\le x\le 1$.