Strong Approximation via Sidon Type Inequalities Original Research Article

The aim of this paper is twofold. First we want to show how a duality relation provides a vehicle to deduce strong summability and approximation properties of Fourier series from some basic inequalities, called Sidon type inequalities. This way the technicalities concerning several strong summability and approximation problems can be reduced to proving such inequalities. On the other hand, we will isolate two properties that induce the sharpest version of these inequalities for a number of orthonormal systems, find their counterparts in terms of strong approximation, and show some of their consequences. We note that these results are known to be the best possible for the trigonometric system