
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/10.18500/1816979120181814961
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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 4961
\mathnet{http://mi.mathnet.ru/isu744}
\crossref{https://doi.org/10.18500/1816979120181814961}
\elib{http://elibrary.ru/item.asp?id=35647730}
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