Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Shcherbakov V. I. Dini – Lipschitz Test on the Generalized Haar Systems. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 4, pp. 435-448. DOI: 10.18500/1816-9791-2016-16-4-435-448, EDN: XHPYIR

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
14.11.2016
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Language: 
Russian
Heading: 
Mathematics
UDC: 
517.52
DOI: 
10.18500/1816-9791-2016-16-4-435-448
EDN: 
XHPYIR

Dini – Lipschitz Test on the Generalized Haar Systems

Autors: 
Shcherbakov Viktor Innokentievich, Moscow Technical University of Communications and Informatics, Russia
Abstract: 

Generalized Haar systems, which are generated (generally speaking, unbounded) by a sequence {pn} ∞n=1 and which is defined on the modification segment [0, 1]∗ , thai is on a segment [0, 1], where {pn} — 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 ωn(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 {pn} ∞n=1 with sup n pn = ∞ a model of the continuous on [0, 1]∗ function, which Fourier Series by generalized Haar’s system, which generated by sequence {pn} ∞n=1 boundly diverges in some fixed point, is constructed. This result may be applied to the zero-dimentions compact Abelian groups. 

Key words: 
Abelian group
modification segment [0; 1]
a continuous functions on the modification segment [0; 1]
characters systems
Price’s systems
a generalized Haar’s systems
Dirichler’s kernels
Dini – Lipschitz’s test
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Received: 
19.07.2016
Accepted: 
27.10.2016
Published: 
30.11.2016
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