Geodesic
Conservative Means of Orthogonal Series and the Spaces $L^p[0;1]$, $p\in (1;\infty )$
Matematičeskie zametki, Tome 71 (2002) no. 2, pp. 182-193.

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

Necessary and sufficient conditions for an orthogonal series to be the Fourier series of a function in the space $L^p[0;1]$, $p\in (1;\infty )$, are obtained. In the special case of regular summation methods we recover the classical results of Orlicz and Lomnicki. 
@article{MZM_2002_71_2_a2,
     author = {I. N. Brui},
     title = {Conservative {Means} of {Orthogonal} {Series} and the {Spaces} $L^p[0;1]$, $p\in (1;\infty )$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {182--193},
     publisher = {mathdoc},
     volume = {71},
     number = {2},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_71_2_a2/}
}
TY  - JOUR
AU  - I. N. Brui
TI  - Conservative Means of Orthogonal Series and the Spaces $L^p[0;1]$, $p\in (1;\infty )$
JO  - Matematičeskie zametki
PY  - 2002
SP  - 182
EP  - 193
VL  - 71
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2002_71_2_a2/
LA  - ru
ID  - MZM_2002_71_2_a2
ER  - 
%0 Journal Article
%A I. N. Brui
%T Conservative Means of Orthogonal Series and the Spaces $L^p[0;1]$, $p\in (1;\infty )$
%J Matematičeskie zametki
%D 2002
%P 182-193
%V 71
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2002_71_2_a2/
%G ru
%F MZM_2002_71_2_a2
I. N. Brui. Conservative Means of Orthogonal Series and the Spaces $L^p[0;1]$, $p\in (1;\infty )$. Matematičeskie zametki, Tome 71 (2002) no. 2, pp. 182-193. http://geodesic.mathdoc.fr/item/MZM_2002_71_2_a2/

[1] Orlicz W., “Beiträge zur Theorie der Orthogonalentwicklungen”, Studia Math., 1 (1929), 1–39

[2] Kachmazh S., Shteingauz G., Teoriya ortogonalnykh ryadov, GIFML, M., 1958

[3] Bari N. K., Trigonometricheskie ryady, GIFML, M., 1961

[4] Zigmund A., Trigonometricheskie ryady, T. 1, Mir, M., 1965

[5] Edvards R., Ryady Fure v sovremennom izlozhenii, T. 1, Mir, M., 1985

[6] Riss F., Sëkefalvi-Nad B., Lektsiya po funktsionalnomu analizu, Mir, M., 1979

[7] Baron S., Vvedenie v teoriyu summiruemosti ryadov, Valgus, Tallin, 1977 | Zbl

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

[9] Brui I. N., “Lineinye srednie ryadov Fure i klassy funktsii”, Matematicheskie issledovaniya, Sb. nauchn. rabot prepodavatelei, aspirantov i studentov matematicheskogo fakulteta Grodnenskogo gos. un-ta. T. 2, Grodno, 1995, 6–11

[10] Brui I. N., “Trigonometricheskie ryady klassov $L^p(\mathbb T)$ i ikh matrichnye srednie”, Vestsi NAN Belarusi. Ser. fiz.-mat. navuk, 2000, no. 1, 46–49 | MR

[11] Brui I. N., “Trigonometricheskie ryady klassov $L^p(\mathbb T)$ i ikh regulyarnye srednie”, Vestsi Akademii navuk Belarusi. Ser. fiz.-mat. navuk, 1996, no. 1, 24–30 | MR | Zbl

[12] Brui I. N., “Trigonometricheskie ryady klassov $L^p(\mathbb T)$, $p\in(1;\infty)$, i ikh konservativnye srednie”, Matem. zametki, 62:5 (1997), 677–686 | MR | Zbl

[13] Khardi G., Raskhodyaschiesya ryady, IL, M., 1951

[14] Fekete M., “Über faktorenfolgen, welche die “Klasse” einer Fourierschen Reihe unverändert lassen”, Acta litterarum ac scientiarum regiae universitatis hungaricae Francisco–Josephinae. Sectio scientiarum mathematicarum, 1922–1923, no. 1, 148–166 | MR

[15] Riesz M., “Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires”, Acta Math., 49 (1926), 465–497 | DOI

[16] Butzer P. L., Nessel R. J., Fourier Analysis and Approximation, V. 1, Birkhäuser Verlag, Basel–Stuttgart, 1971 | MR

[17] Edvards R., Ryady Fure v sovremennom izlozhenii, T. 2, Mir, M., 1985

[18] Büntinas M., “Some new multipliers of Fourier series”, Proc. Amer. Math. Soc., 101:3 (1987), 497–502 | DOI | MR

[19] Giang D. V., Móricz F., “Multipliers of Fourier transforms and series on $L^1$”, Archiv der Mathematik, 62:3 (1994), 230–238 | DOI | MR | Zbl

[20] Zigmund A., Trigonometricheskie ryady, T. 2, Mir, M., 1965

[21] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969

[22] Tynnov M., “$T$-dopolnitelnye prostranstva koeffitsientov Fure”, Uch. zapiski Tartuskogo gos. un-ta, 192 (1966), 65–81 | MR | Zbl

[23] Tynnov M., “Koeffitsienty Fure i mnozhiteli summiruemosti”, Uch. zapiski Tartuskogo gos. un-ta, 253 (1970), 194–201 | MR | Zbl

[24] Rooney P. G., “On the representation of sequence as Fourier coefficients”, Proc. Amer. Math. Soc., 11:5 (1960), 762–768 | DOI | MR | Zbl

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

[26] Torchinsky A., Real-Variable Methods in Harmonic Analysis, Academic Press, Orlando–San Diego–New York, 1986 | Zbl

[27] Stein I. M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973

[28] Nursultanov E. D., “O multiplikatorakh ryadov Fure po trigonometricheskoi sisteme”, Matem. zametki, 63:2 (1998), 235–247 | MR | Zbl

[29] Bruj I., Schmieder G., “Real trigonometric series of class BMO and $(C,1)$-means”, Acta Sci. Math. (Szeged), 64 (1998), 483–488 | MR | Zbl