# Article

Full entry | PDF   (0.2 MB)
Keywords:
Kurzweil-Henstock integral; derivation basis; locally compact zero-dimensional abelian group; characters of a group; multiplicative integral transform; inversion formula.
Summary:
A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case.
References:
[1] Agaev, G. N., Vilenkin, N. Ya., Dzhafarli, G. M., Rubistein, A. I.: Multiplicative system of functions and harmonic analysis on zero-dimensional groups. Baku (1981), Russian.
[2] Golubov, B., Efimov, A., Skvortsov, V.: Walsh Series and Transforms: Theory and Applications. Kluwer Academic Publishers (1991). MR 1155844 | Zbl 0785.42010
[3] Guzman, M. de: Differentiation of Integral in $\mathbb R^n$. Springer-Verlag, Berlin (1975). MR 0457661
[4] Lee, P. Y., Výborný, R.: The Integral: An Easy Approach after Kurzweil and Henstock. Austral. Math. Soc. Lectures Series 14, University Press, Cambridge (2000). MR 1756319
[5] Ostaszewski, K. M.: Henstock integration in the plane. Memoirs of the AMS, Providence 63 (1986). MR 0856159 | Zbl 0596.26005
[6] Skvortsov, V. A., Tulone, F.: Henstock type integral in harmonic analysis on zero-dimensional groups. J. Math. Anal. Appl. 322 (2006), 621-628. DOI 10.1016/j.jmaa.2005.09.053 | MR 2250603 | Zbl 1129.43300
[7] Skvortsov, V. A., Tulone, F.: $\mathcal P$-adic Henstock integral in the theory of series with respect to characters of zero-dimensional groups. Vestnik Moskov. Gos. Univ. Ser. Mat. Mekh. 1 (2006), 25-29 Engl. transl. Moscow Univ. Math. Bull. 61 (2006), 27-31. MR 2255051
[8] Thomson, B. S.: Derivation bases on the real line. Real Anal. Exchange 8 (1982/83), 67-207 and 278-442. DOI 10.2307/44151585
[9] Thomson, B. S.: Derivates of interval functions. Memoirs of the AMS, Providence 93 (1991). MR 1078198 | Zbl 0734.26003

Partner of