Abstract
Strong approximation of Fourier series on generalized periodic Morrey spacesIn recent years, a lot of attention has been paid to study of Morrey type spaces.Many applications in partial differential equation of Morrey spaces and Lizorkin-Triebel spaces have been given in work G.Di Fazioand, M. Ragusa and the book of T. Mizuhara.The theory of generalized Triebel-Lizorkin-Morrey spaces is developed.Generalized Morrey spaces, with T. Mizuhara and E. Nakai proposed, are equipped with a parameter and a function.First we give definition of Morrey and generalized Morrey spaces.Then we recall the boundedness of periodic Hilbert transform.This will be our main tool for all wtht follows.In a more or less elementary way, we carry over the known boundedness assertions for the Hilbert transform on Morrey spaces defined on R to periodic Morrey spaces.Boundedness of the Hilbert transform implies uniform estimates of the operator norms of the partial sumd of the Fourier series.Then we study vectorvalued Fourier-multiplier theorem for smooth multipliers.Afterwards, we study vector valued version of famous Riesz theorem.Here we concentrate on Lizorkin representations.Finally, we get an interesting characterization of the space E s ϕ,p,q (T ) by using differences of partial sums of the Fourier series Finally, we get an interesting characterization of the space E s ϕ,p,q (T ) by using differences of partial sums of the Fourier series and consequence for strong approximation of Fourier series on Morrey space.