A note on Grüss-type inequalities for positive linear operators

Grüss-type inequalities form an important class of inequalities in approximation theory. A Grüss-type inequality for sequences of positive linear operators on C[a, b] is already known in the literature. We obtain a trigonometric version of this inequality for positive linear operators acting on the space of all continuous \(2\pi \) -periodic functions, \(C_{2\pi }(\mathbb {R})\) . We also prove a restricted theorem for the subspace of even \(2\pi \) -periodic continuous functions. We illustrate these theorems on positive linear operators that are connected with preconditioning problem of Toeplitz matrices. In particular, we apply it to positive linear operators connected to circulant, Hartley and \(\tau \) - preconditioners.