On the norm of the Fourier-Gegenbauer projection in weighted \(L_p\) spaces (Q1299984)

Let \(p^\lambda_k\), \(k= 0,1,\dots\), be the normalized Gegenbauer polynomials orthogonal with respect to \(w^\lambda= (1- x^2)^{\lambda- 1/2}\) on \([-1,1]\), and consider the Fourier projection \[ S_nf= \sum^n_{k= 0}a_k(f) p^\lambda_k. \] The authors give upper and lower bounds for \(\|S_n\|^\lambda_p\) for \(p\geq (2\lambda+ 1)/\lambda\) or \(1<p\leq (2\lambda+ 1)/(\lambda+ 1)\). In particular, they show that \(\|S_n\|^\lambda_p\) is unbounded at the critical indices \((2\lambda+ 1)/(\lambda+ 1)\) and \((2\lambda+ 1)/\lambda\).