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Navasardyan K. A. On the Representation of Functions by Absolutely Convergent Series by H-system. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2018, vol. 18, iss. 1, pp. 49-61. DOI: 10.18500/1816-9791-2018-18-1-49-61, EDN: LDSCQS
On the Representation of Functions by Absolutely Convergent Series by H-system
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 (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 H, which converges to f a.e. on X . From this theorem, in particular, it follows that if 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 H will converge uniformly. The paper results are obtained using the methods of metrical functions theory.
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