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

K. A. Navasardyan

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 1, pp. 49-61

Résumé

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.