Geodesic
On the representation of functions by absolutely convergent series by $\mathcal{H}$-system
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 1, pp. 49-61 
K. A. Navasardyan. On the representation of functions by absolutely convergent series by $\mathcal{H}$-system. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 1, pp. 49-61. http://geodesic.mathdoc.fr/item/ISU_2018_18_1_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Macias R., Segovia C., “Lipschitz functions on spaces of homogeneous type”, Adv. in Math., 33 (1979), 271–309 | DOI | MR

[2] Christ A., “A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral”, Colloquium Math., 60/61:2 (1990), 601–628 | DOI | MR

[3] Aimar H., Bernardis A., Iaffel B., “Multiresolution Approximations and Unconditional Bases on Weighted Lebesgue Spaces on Spaces of Homogeneous Type”, J. Approx. Theory, 148:1 (2007), 12–34 | DOI | MR

[4] Aimar H., Bernardis A., Nowak L., “Equivalence of Haar Bases Associated to Different Dyadic Systems”, J. of Geometric Analysis, 21:2 (2011), 288–304 | DOI | MR

[5] Aimar H., Bernardis A., Nowak L., “Dyadic Fefferman–Stein Inequalities and the Equivalence of Haar Bases on Weighted Lebesgue Spaces”, Proceedings of the Royal Society of Edinburgh Section A: Math., 141:1 (2011), 1–22 | DOI | MR

[6] Luzin N. N., Integral and trigonometric series, Gostehizdat, M.–L., 1951, 550 pp. (in Russian) | MR

[7] Arutyunyan F. G., “On series in the Haar system”, Dokl. Akad. Nauk Armyanskoi SSR, 42:3 (1966), 134–140 (in Russian) | MR

[8] Davtyan R. S., “On Representation of Functions by Orthogonal Series Possessing Martingale Properties”, Math. Notes, 19:5 (1976), 405–409 | DOI | MR

[9] Gevorkjan G. G., “On the Representation of Measurable Functions by Martingales”, Analysis Math., 8:4 (1982), 239–256 | DOI | MR

[10] Gevorkian G. G., “Representation of Measurable Functions by Absolutely Convergent Series of Translates and Dilates of One Function”, East J. Approx., 2:4 (1996), 439–458 | MR

[11] Golubov B. I., “On a class of complete orthogonal systems”, Sibirskij matematiceskij zurnal, 9:2 (1968), 297–314 (in Russian) | MR