Jordan test for the Haar-type systems

We consider Haar-type systems, which are generated by a (generally speaking, unbounded) sequence $ \{ p_n \}_{n=1}^\infty $, and which are defined on the modified segment $ [0, 1]^* $, i.\,e., on the segment [0, 1] whose $ \{ p_n \}$-rational points are calculated two times. The main result of this work is a Jordan-type test for the pointwise and uniform convergence of Fourier series with respect to Haar-type systems. It is shown that the test obtained in the paper can not be improved. An example of a function of bounded variation, whose Fourier series with respect to Haar-type system diverges at some point, is constructed in the case of $ \sup\limits_n p_n = \infty$. Thus for any unbounded sequence $ \{ p_n \}_{n=1}^\infty $ there exists a monotone function whose Fourier series with respect to Haar-type system, generated by the given sequence $ \{ p_n \}_{n=1}^\infty $, diverges at some point. It is found that the Jordan test of convergence of Fourier series with respect to Haar-type systems does not differ from the Dini--Lipschitz condition for those systems. As the Dini--Lipschitz condition was considered earlier, the main value of this work is the construction of a corresponding counterexample, i.\,e., the construction of an example (a model) of a function of bounded variation whose Fourier series with respect to Haar-type Systems diverges at some point. In the counterexamples from the earlier works on the Dini--Lipschitz criterion (as well as for all Dini convergence criteria), the functions were {\it not of bounded variation}. The article's conclusion discusses how the Jordan convergence criterion evolved when transitioning from trigonometric systems of functions to Price systems (and to N.Ya. Vilenkin systems), and from there to generalized Haar systems and Haar-type systems.

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