George Tephnadze
abstract:
The aim of this monograph is to discuss, develop and apply the newest
developments of this fascinating theory connected with modern harmonic analysis.
In particular, we investigate Fejér kernels
and study two-sided estimates for them, then we prove the almost everywhere
convergence of Fejér means. Moreover, we
define the Lebesgue and Walsh-Lebesgue points of integrable functions and prove
the convergence of Fejér means of integrable
functions with respect to the Walsh system at Walsh-Lebesgue points, which are
the almost everywhere points for any integrable function. We also study the
convergence of Fejér means in $L_p$ norm.
Moreover, we investigate some important upper and lower estimates of Fejér
kernels that are very crucial in proving the important results in the theory of
martingale Hardy spaces. In particular, we consider the maximal operator,
restricted maximal operators and weighted maximal operators of Fejér
means with respect to the Walsh system and prove the $(H_p,L_p)$ and $(H_p,\textit{weak-}L_p)$
type inequalities for them. Next, we apply these results to find the necessary
and sufficient conditions for the modulus of continuity under which the
convergence of norm Fejér means occurs. We
also study analogous problems for the subsequences of Fejér
means in terms of modulus of continuity. Finally, we prove some new Hardy type
inequalities for Fejér means, which are
called strong convergence results of Fejér
means. We also prove the sharpness of all our main results in the present
monograph.
Mathematics Subject Classification: 42C10
Key words and phrases: Dyadic group, Walsh system, $L_{p}$ space, \textit{weak}-$L_{p}$ space, modulus of continuity, Fejér, dyadic martingale, martingale Hardy spaces, maximal operators, restricted maximal operators, weighted maximal operators, strong convergence