On the Representation of Functions by Absolutely Convergent Series by H-system

The paper dealswith 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.