Explicit form of the Dirichlet kernel for Walsh series and transformations

The dynamics of the behaviour of the absolute values of the Dirichlet kernels is described: the discrete dynamics for the Dirichlet kernel of the Walsh system and the continuous dynamics for the generalized Walsh–Dirichlet kernel. Estimates of the $p$-norms of the Dirichlet kernels are obtained. The concept of generalized Lebesgue constant is introduced and the corresponding formulae are found, which generalize Fine's formulae for the Lebesgue constants. These results hold not only for the Walsh system in the Paley enumeration, but also for rearrangements of the Walsh systems, including linear and piecewise linear ones.