Integrability of series with respect to multiplicative systems and generalized derivatives

N. Yu. Agafonova ; S. S. Volosivets

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2024), pp. 3-14

Résumé

We give some necessary and sufficient conditions for the convergence of generalized derivatives of sums of series with respect to multiplicative systems and the corresponding Fourier series. These conditions are counterparts of trigonometric results of S. Sheng, W.O. Bray and $\check{C}$.V. Stanojević and extend some results of F. Móricz proved for Walsh–Fourier series.

Détail

Keywords: multiplicative system, $L^1$-integrability, generalized derivative.
Mots-clés : $L^1$-convergence

Bibliographie

[1] Stanojević Č.V., Stanojević V.B., “Generalizations of the Sidon–Telyakovskii theorem”, Proc. Amer. Math. Soc., 101:4 (1987), 679–684 | MR | Zbl

[2] Telyakovskii S.A., “Ob odnom dostatochnom uslovii Sidona integriruemosti trigonometricheskikh ryadov”, Matem. zametki, 14:3 (1973), 317–328 | MR | Zbl

[3] Sheng S., “The extension of the theorems of C.V. Stanojević and V.B. Stanojević”, Proc. Amer. Math. Soc., 110:4 (1990), 895–904 | DOI | MR | Zbl

[4] Bray W.O., Stanojević Č.V., “Tauberian $L^1$-convergence classes of Fourier series”, Trans. Amer. Math. Soc., 275:1 (1983), 59–69 | MR | Zbl

[5] Móricz F., “Sidon-type inequalities”, Publ. Inst. Math. (Beograd), 48(62) (1990), 101–109 | MR | Zbl

[6] Golubov B.I., Efimov A.V., Skvortsov V.A., Ryady i preobrazovaniya Uolsha. Teoriya i primeneniya, Nauka, M., 1987 | MR

[7] Agaev G.N., Vilenkin N.Ya., Dzhafarli G.M., Rubinshtein A.I., Multiplikativnye sistemy funktsii i garmonicheskii analiz na nulmernykh gruppakh, Elm, Baku, 1981 | MR

[8] Vilenkin N.Ya., “Ob odnom klasse polnykh ortonormalnykh sistem”, Izv. AN SSSR. Ser. matem., 11:4 (1947), 363–400 | Zbl

[9] Pal J., Simon P., “On a generalization of the concept of derivative”, Acta Math. Acad. Sci. Hungar., 29:1-2 (1977), 155–164 | DOI | MR | Zbl

[10] Volosivets S.S., “Priblizhenie funktsii ogranichennoi $p$-fluktuatsii polinomami po multiplikativnym sistemam”, Anal. Math., 21:1 (1995), 61–77 | DOI | MR | Zbl

[11] Schipp F., Wade W.R., Simon P., Walsh series. An introduction to dyadic analysis, Akad. Kiado, Budapest, 1990 | MR

[12] Volosivets S.S., Likhacheva T.V., “Neravenstva tipa Sidona i silnaya approksimatsiya summami Fure po multiplikativnym sistemam”, Sib. matem. zhurn., 57:3 (2016), 617–631 | MR | Zbl

[13] Móricz F., “On $L^1$-convergence of Walsh–Fourier series. II”, Acta Math. Acad. Sci. Hungar., 58:1-2 (1991), 203–210 | DOI | MR | Zbl

[14] Móricz F., “On $L^1$-convergence of Walsh–Fourier series. I”, Rend. Circ. Matem. Palermo. Ser. 2, 38:3 (1989), 411–418 | DOI | MR | Zbl

[15] Stanojević Č.V., “Classes of $L^1$-convergence of Fourier and Fourier–Stieltjes series”, Proc. Amer. Math. Soc., 82:2 (1981), 209–215 | MR | Zbl