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Multipliers in the Hardy spaces $H_p(D^m)$ with $p\in (0,1]$ and approximation properties of summability methods for power series
Sbornik. Mathematics, Tome 188 (1997) no. 4, pp. 621-638 Cet article a éte moissonné depuis la source Math-Net.Ru

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For $p\in (0,1]$ conditions on a number sequence $\{\lambda _k\}_0^\infty$ are indicated ensuring that the multiplier operator $\sum _{k=0}^\infty c_k z^k \mapsto \sum _{k=0}^\infty \lambda _k c_k z^k$ is continuous in the Hardy space $H_p(D)$ (here $D$ can also be a polydisc $D^m$). Some sufficient conditions are also established. These results are used to find out the precise order of approximation of multiple power series by Bochner–Riesz means and to evaluate the $K$-functional for a pair of spaces related to the polyharmonic operator. 
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R. M. Trigub. Multipliers in the Hardy spaces $H_p(D^m)$ with $p\in (0,1]$ and approximation properties of summability methods for power series. Sbornik. Mathematics, Tome 188 (1997) no. 4, pp. 621-638. http://geodesic.mathdoc.fr/item/SM_1997_188_4_a4/

[1] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[2] Rudin U., Teoriya funktsii v polikruge, Mir, M., 1974 | MR | Zbl

[3] Trigub R. M., “Absolyutnaya skhodimost integralov Fure, summiruemost ryadov Fure i priblizhenie polinomami funktsii na tore”, Izv. AN SSSR. Ser. matem., 44:6 (1980), 1378–1409 | MR | Zbl

[4] Trigub R. M., “Formula dlya $K$-funktsionala pary prostranstv funktsii neskolkikh peremennykh”, Issledovaniya po teorii funktsii mnogikh peremennykh, Mezhvuz. sb., Izd-vo Yarosl. gos. un-ta, Yaroslavl, 1988, 122–127 | MR

[5] Trigub R. M., “Multiplikatory ryadov Fure”, Ukr. matem. zhurn., 1991, no. 12, 1686–1693 | MR | Zbl

[6] Storozhenko E. A., “O teoremakh tipa Dzheksona v $H^p$, $0

1$”, Izv. AN SSSR. Ser. matem., 44:4 (1980), 946–962 | MR | Zbl

[7] Valashek Ya., “O priblizhenii v mnogomernykh prostranstvakh Khardi $H^p$, $0

\leqslant 1$”, Soobsch. AN Gr. SSR, 105:1 (1982), 21–24 | MR | Zbl

[8] Storozhenko E. A., Krotov V. G., “Approksimativnye i differentsialno-raznostnye svoistva funktsii iz prostranstv Khardi $H^p$, $0

1$”, Teoriya funktsii i priblizhenii, Mezhvuz. sb., Trudy Saratovskoi zimnei shkoly, ch. I (25 yanv.–5 fevr. 1982 g.), Izd-vo Sarat. un-ta, Saratov, 1983, 65–80 | MR

[9] Oswald P., “On Some Approximation Properties of Real Hardy Spaces $(0

\leqslant 1)$”, J. Approx. Theory, 40:1 (1984), 45–65 | DOI | MR | Zbl

[10] Solyanik A. A., “On the order of approximation to functions of $H^p(\mathbb R)$ $(0

\leqslant 1)$ by certain means of Fourier integrals”, Anal. Math., 12 (1986), 59–75 | DOI | MR

[11] Colzani L., “Jackson Theorems in Hardy Spaces and Approximation by Riesz Means”, J. Approx. Theory, 49 (1987), 240–251 | DOI | MR | Zbl

[12] Belinskii E. S., “Silnaya summiruemost periodicheskikh funktsii i teoremy vlozheniya”, DAN, 332:2 (1993), 133–134 | MR | Zbl

[13] Pribegin S. G., Otsenki sverkhu i snizu skorosti priblizheniya funktsii klassa $H^p$, $0

\leqslant 1$ srednimi Rissa, Dep. v GNTB Ukr. 29.06.93 No. 1290-Uk.93

[14] Trigub R. M., “Multiplikatory v prostranstvakh Khardi $H_p(D^m)$ pri $p\in(0,1]$ i approksimativnye svoistva metodov summirovaniya stepennykh ryadov”, DAN, 335:6 (1994), 697–699 | MR | Zbl

[15] Brudnyi Yu. A., Krein S. G., Semenov E. M., “Interpolyatsiya lineinykh operatorov”, Itogi nauki i tekhniki. Matem. analiz, VINITI, M., 1986, 3–163 | MR

[16] Arestov V. V., “Ob integralnykh neravenstvakh dlya trigonometricheskikh polinomov i ikh proizvodnykh”, Izv. AN SSSR. Ser. matem., 45:1 (1981), 3–22 | MR | Zbl

[17] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[18] Trigub R. M., “Integriruemost i asimptoticheskoe povedenie preobrazovaniya Fure radialnoi funktsii”, Metricheskie voprosy teorii funktsii i otobrazhenii, Naukova dumka, Kiev, 1977, 142–163 | MR

[19] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR

[20] Trigub R. M., “Multiplikatory Fure i nekotorye voprosy teorii priblizheniya”, Internat. Conf. on Appr. Theory dedicated' to the memory of Prof. P. P. Korovkin, Abstracts, v. 2 (Kaluga, June 26–29, 1996), 214–216