On two-sided estimate for norm of Fourier operator

Abstract

In the work we study the behavior of Lebesgue constant $L_n$ of the Fourier operator defined in the space of continuous $2\pi$-periodic functions. The known integral representations expressed in terms of the improper integrals are too cumbersome. They are complicated both for theoretical and practical purposes.We obtain a new integral representation for $L_n$ as a sum of Riemann integrals defined on bounded converging domains. We establish equivalent integral representations and provide strict two-sided estimates for their components. Then we provide a two-sided estimate for the Lebesgue constant. We solve completely the problem on the upper bound of the constant $L_n$. We improve its known lower bound.