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 Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.: Year: Volume: Issue: Page: Find

 Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2018, Volume 18, Issue 1, Pages 49–61 (Mi isu744)

Scientific Part
Mathematics

On the representation of functions by absolutely convergent series by $\mathcal{H}$-system

K. A. Navasardyan

Yerevan State University, 1, Alex Manoogian Str., Yerevan, Republic of Armenia, 0025

Abstract: The paper deals with the representation of absolutely convergent series of functions in spaces of homogeneous type. The definition of a system of Haar type ($\mathcal{H}$-system) associated to a dyadic family on a space of homogeneous type X is given in the Introduction. It is proved that for almost everywhere (a.e.) finite and measurable on a set $X$ function $f$ there exists an absolutely convergent series by the system $\mathcal {H}$, which converges to $f$ a.e. on $X$. From this theorem, in particular, it follows that if $\mathcal{H} = \{h_n \}$ is a generalized Haar system generated by a bounded sequence $\{p_k\}$, then for any a.e. finite on $[0,1]$ and measurable function $f$ there exists an absolutely convergent series in the system $\{h_n \}$, which converges a.e. to $f (x)$. It is also proved, that if $X$ is a bounded set, then one can change the values of an a.e. finite and measurable function on a set of arbitrary small measure such that the Fourier series of the obtained function with respect to system $\mathcal{H}$ will converge uniformly. The paper results are obtained using the methods of metrical functions theory.

Key words: Haar type system, dyadic family, absolute convergence, uniform convergence.

DOI: https://doi.org/10.18500/1816-9791-2018-18-1-49-61

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UDC: 517.51

Citation: K. A. Navasardyan, “On the representation of functions by absolutely convergent series by $\mathcal{H}$-system”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 18:1 (2018), 49–61

Citation in format AMSBIB
\Bibitem{Nav18} \by K.~A.~Navasardyan \paper On the representation of functions by absolutely convergent series by $\mathcal{H}$-system \jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform. \yr 2018 \vol 18 \issue 1 \pages 49--61 \mathnet{http://mi.mathnet.ru/isu744} \crossref{https://doi.org/10.18500/1816-9791-2018-18-1-49-61} \elib{http://elibrary.ru/item.asp?id=35647730}