The boundedness of (sub)sequences of partial Fourier and Fourier-Walsh sums in subspaces of separable Orlicz spaces is studied. The boundedness of the shift operator and Paley function with respect to the Haar system is also investigated. These results are applied to get the analogues of the classical theorems on basicness of the trigonometric and Walsh systems in nonreflexive separable Orlicz spaces.
@article{bwmeta1.element.bwnjournal-article-smv121i2p193bwm,
author = {C. Finet and G. Tkebuchava},
title = {Some classical function systems in separable Orlicz spaces},
journal = {Studia Mathematica},
volume = {119},
year = {1996},
pages = {193-205},
zbl = {0861.42020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv121i2p193bwm}
}
Finet, C.; Tkebuchava, G. Some classical function systems in separable Orlicz spaces. Studia Mathematica, Tome 119 (1996) pp. 193-205. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv121i2p193bwm/
[00000] [1] Bari N. K., and Stechkin S. B., Best approximation and differential properties of two conjugate functions, Trudy Moskov. Mat. Obshch. 5 (1956), 483-522 (in Russian).
[00001] [2] Bloom S., and Kerman R., Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal function, Studia Math. 110 (1994), 149-167. | Zbl 0813.42014
[00002] [3] Burkholder D., Distribution function inequalities for martingales, Ann. Probab. 11 (1973), 19-42. | Zbl 0301.60035
[00003] [4] Burkholder D., Davis B., and Gundy R., Integral inequalities for convex functions of operators on martingales, in: Proc. Sixth Berkeley Sympos. Math. Statist. Probab., Vol. II, Univ. of California Press, 1972, 223-240. | Zbl 0253.60056
[00004] [5] Ciesielski Z., and Kwapień S., Some properties of the Haar, Walsh-Paley, Franklin and the bounded polygonal orthonormal bases in spaces, Comment. Math., Tomus Specialis in Honorem L. Orlicz, 1979, part 2, 37-42. | Zbl 0433.46012
[00005] [6] Fridli S., Ivanov V., and Simon P., Representation of functions in the space φ(L) by Vilenkin series, Acta Sci. Math. (Szeged) 48 (1985), 143-154. | Zbl 0589.42018
[00006] [7] Gogatishvili A., Kokilashvili V., and Krbec M., Maximal functions, φ(L) classes and Carleson measures, Proc. A. Razmadze Math. Inst. 102 (1993), 85-97. | Zbl 0815.42010
[00007] [8] Kashin B. S., and Saakjan A. A., Orthogonal Series, Amer. Math. Soc., Providence, 1989.
[00008] [9] Kolmogoroff A. N., Sur les fonctions harmoniques conjuguées et les séries de Fourier, Fund. Math. 7 (1927), 25-28.
[00009] [10] Komissarov A. A., Equivalence of Haar and Franklin systems in some function spaces, Sibirsk. Mat. Zh. 23 (5) (1982), 115-126 (in Russian). | Zbl 0508.46017
[00010] [11] Konyagin S. V., On subsequences of partial Fourier-Walsh sums, Mat. Zametki 54 (4) (1993), 69-75 (in Russian).
[00011] [12] Krasnosel'skiĭ M. A., and Rutickiĭ Ya. B., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
[00012] [13] Lozinski S., On convergence and summability of Fourier series and interpolation processes, Mat. Sb. 14 (3) (1944), 175-268. | Zbl 0060.18301
[00013] [14] Paley R. E. A. C., A remarkable system of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279. | Zbl 0005.24901
[00014] [15] Riesz M., Sur les fonctions conjuguées, Math. Z. 27 (1927), 218-244. | Zbl 53.0259.02
[00015] [16] Ryan R., Conjugate functions in Orlicz spaces, Pacific J. Math. 13 (1963), 1371-1377. | Zbl 0133.37301
[00016] [17] Tkebuchava G. E., On unconditional bases in nonreflexive function spaces, Soobshch. Akad. Nauk Gruzin. SSR 101 (2) (1981), 297-299 (in Russian); English transl. in: Fifteen Papers on Functional Analysis, Amer. Math. Soc. Transl. (2) 124, 1984, 17-19. | Zbl 0455.46026
[00017] [18] Tsereteli O. P., On the integrability of conjugate functions, Trudy Tbilissk. Mat. Inst. Razmadze Akad. Nauk. Gruzin. SSR 45 (1973), 149-168 (in Russian).
[00018] [19] Watari C., Mean convergence of Walsh-Fourier series, Tôhoku Math. J. 16 (1964), 183-188. | Zbl 0146.08901
[00019] [20] Zygmund A., Trigonometric Series, Vols. I, II, Cambridge Univ. Press, 1968.