Abstract
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 supp n < ∞. 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 supp n = ∞.
Similar content being viewed by others
REFERENCES
V. A. Skvortsov, “Null-series with respect to some multiplicative system,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1979), no. 6, 63–67.
K. Yoneda, “On generalized A-integrals. I,” Proc. Japan Acad., 45 (1969), no. 3.
K. Yoneda, “On generalized A-integrals. II,” Math. Japon., 18 (1973), no. 2.
V. V. Kostin and V. A. Skvortsov, “Martingale sequences in the theory of orthogonal series,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1999), no. 6, 50–53.
A. N. Shiryaev, Probability [in Russian], Nauka, Moscow, 1989.
G. G. Gevorkyan, “The majorant and uniqueness of series with respect to the Franklin system,” Mat. Zametki [Math. Notes], 59 (1996), no. 4, 521–545.
G. G. Gevorkyan, “Uniqueness of series with respect to the Franklin system,” Mat. Zametki [Math. Notes], 42 (1989), no. 2, 51–58.
B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow, 1987.
R. F. Gundy, “Martingale theory and positive convergence of certain orthogonal series,” Trans. Amer. Math. Soc., 124 (1966), no. 2, 228–248.
Y. S. Chow, “Convergence theorems of martingales,” Z. Wahrscheinlichkeitstheorie, 1 (1963), 340–346.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kostin, V.V. Reconstructing Coefficients of Series from Certain Orthogonal Systems of Functions. Mathematical Notes 73, 662–679 (2003). https://doi.org/10.1023/A:1024012705318
Issue Date:
DOI: https://doi.org/10.1023/A:1024012705318