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.
UDC:
517.51
DOI:
10.18500/1816-9791-2018-18-1-49-61
Fulltext:
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