Dini–Lipschitz test on the generalized Haar systems

Generalized Haar systems, which are generated (generally speaking, unbounded) by a sequence $ \{p_n\}_{n=1}^\infty $ and which is defined on the modification segment $ [0, 1]^* $, thai is on a segment $[0, 1]$, where $ \{ p_n \}$ — rational points are calculated two times and which is a geometrical representation of zero-dimensional compact Abelians group are considering in this work. The main result of this work is a setting of the pointwise estimation between of an absolute value of difference between continuous in the given point function and it's $n$-s particular Fourier sums and “pointwise” module of continuity of this function (this notion (“pointwise” module of continuity $\omega_n (x, f)$) is also defined in this work). Based on this a uniform estimation between an absolute value of difference between a continious on the $ [0, 1]^* $ function and it's particular Fourier Sums and the module of continuity of this function is established. A sufficient condition of the pointwise and uniformly boundedness of particular Fourier Sums by generalized Haar's systems for the given continuous function is established too. Based on this estimation we establish a test of convergence of Fourier Series with respect to generalized Haar's systems analogous Dini–Lipschitz test. The unimprovement of the test, which is obtained in this work, is showed too. For any $ \{ p_n \}_{n=1}^\infty $ with $ \sup\limits_n p_n = \infty $ a model of the continuous on $ [0, 1]^* $ function, which Fourier Series by generalized Haar's system, which generated by sequence $ \{ p_n \}_{n=1}^\infty $ boundly diverges in some fixed point, is constructed. This result may be applied to the zero-dimentions compact Abelian groups.