Geodesic
Reconstructing Coefficients of Series from Certain Orthogonal Systems of Functions
Matematičeskie zametki, Tome 73 (2003) no. 5, pp. 704-723.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a series with respect to a multiplicative Price system or a generalized Haar system and assume that the martingale subsequence of its partial sums converges almost everywhere. In this paper we prove that, under certain conditions imposed on the majorant of this sequence, the series is a Fourier series in the sense of the $A$-integral (or its generalizations) of the limit function if the series is considered as a series with respect to a system with $\sup p_n\infty$. In similar terms, we also present sufficient conditions for a series to be a Fourier series in the sense of the usual Lebesgue integral. We give an example showing that the corresponding assertions do not hold if $\sup p_n=\infty$. 
@article{MZM_2003_73_5_a7,
     author = {V. V. Kostin},
     title = {Reconstructing {Coefficients} of {Series} from {Certain} {Orthogonal} {Systems} of {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {704--723},
     publisher = {mathdoc},
     volume = {73},
     number = {5},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_73_5_a7/}
}
TY  - JOUR
AU  - V. V. Kostin
TI  - Reconstructing Coefficients of Series from Certain Orthogonal Systems of Functions
JO  - Matematičeskie zametki
PY  - 2003
SP  - 704
EP  - 723
VL  - 73
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2003_73_5_a7/
LA  - ru
ID  - MZM_2003_73_5_a7
ER  - 
%0 Journal Article
%A V. V. Kostin
%T Reconstructing Coefficients of Series from Certain Orthogonal Systems of Functions
%J Matematičeskie zametki
%D 2003
%P 704-723
%V 73
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2003_73_5_a7/
%G ru
%F MZM_2003_73_5_a7
V. V. Kostin. Reconstructing Coefficients of Series from Certain Orthogonal Systems of Functions. Matematičeskie zametki, Tome 73 (2003) no. 5, pp. 704-723. http://geodesic.mathdoc.fr/item/MZM_2003_73_5_a7/

[1] Skvortsov V. A., “O nul-ryadakh po nekotoroi multiplikativnoi sisteme”, Vestn. MGU. Ser. 1. Matem., mekh., 1979, no. 6, 63–67 | MR | Zbl

[2] Yoneda K., “On generalized $A$-integrals, I”, Proc. Japan Acad., 45:3 (1969) | DOI | MR | Zbl

[3] Yoneda K., “On generalized $A$-integrals, II”, Math. Japon., 18:2 (1973)

[4] Kostin V. V., Skvortsov V. A., “Martingalnye posledovatelnosti v teorii ortogonalnykh ryadov”, Vestn. MGU. Ser. 1. Matem., mekh., 1999, no. 6, 50–53 | MR | Zbl

[5] Shiryaev A. N., Veroyatnost, Nauka, M., 1989

[6] Gevorkyan G. G., “Mazhoranta i edinstvennost ryadov po sisteme Franklina”, Matem. zametki, 59:4 (1996), 521–545 | MR | Zbl

[7] Gevorkyan G. G., “O edinstvennosti ryadov po sisteme Franklina”, Matem. zametki, 46:2 (1989), 51–58 | MR | Zbl

[8] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[9] Gundy R. F., “Martingale theory and positive convergence of certain orthogonal series”, Trans. Amer. Math. Soc., 124:2 (1966), 228–248 | DOI | MR | Zbl

[10] Chow Y. S., “Convergence theorems of martingales”, Z. Wahrscheinlichkeitstheorie, 1 (1963), 340–346 | DOI | Zbl