A Cohen type inequality for Fourier expansions of orthogonal polynomials with a non‐discrete Gegenbauer‐Sobolev inner product

Let d μ( x ) = (1 − x 2 ) α−1/2 dx ,α> − 1/2, be the Gegenbauer measure on the interval [ − 1, 1] and introduce the non‐discrete Sobolev inner product\documentclass{article}\begin{document}\pagestyle{empty}$$ \langle f,g\rangle = \int_{-1}^{1}f(x) g(x)\, d\mu (x)+ \lambda \int_{-1}^{1}f^{\prime }(x) g^{\prime }(x)\, d\mu (x) $$\end{document} where λ>0. In this paper we will prove a Cohen type inequality for Fourier expansions in terms of the polynomials orthogonal with respect to the above inner product. Results on divergence for Cesàro means of Gegenbauer‐Sobolev expansions are deduced. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim