Majorants of the Dirichlet Kernels and the Dini Pointwise Tests for Generalized Haar Systems

In this paper, we obtain five tests (three of which are symmetric) of pointwise convergence of Fourier series with respect to generalized Haar systems; the tests are similar to the Dini convergence tests. It is shown that the Dini convergence tests for Price systems are also valid for generalized Haar systems. It is also shown that the classical Dini convergence test does not apply, in general, even to generalized Haar systems, although the classical symmetric Dini test for generalized Haar systems is valid. Also upper bounds for the Dirichlet kernels for generalized Haar systems are obtained.