A Cohen-Type Inequality for Jacobi-Sobolev Expansions

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

“…In a well‐known paper 3 Cohen proved that for any trigonometric polynomial $P_N(x)=\sum_{k=1}^{N} a_k e^{in_kx}, $ where 0 < n 1 < … < n N , N ⩾ 2, and | a k | ⩾ 1 for 1 ⩽ k ⩽ N , the following inequality holds: Motivated by the work of Cohen, inequalities of this type have been established in various other contexts, e.g., for classical orthogonal expansions or on compact groups (see 4, 5, 10, 11, 14). Recently, a Cohen type inequality for Fourier expansions with respect to some discrete Sobolev type inner products was done by the author and F. Marcellán (see 8, 9). The main purpose of this article is to prove a Cohen type inequality for Fourier expansions in terms of the polynomials associated to the non‐discrete inner product where λ>0 and d μ( x ) = (1 − x 2 ) α−1/2 dx with α> − 1/2.…”