Download PDF - Some classical function systems in separable Orlicz spacesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Some classical function systems in separable Orlicz spaces

Authors 1, 2

Affiliations

  1. Université de Mons-Hainaut, Institut de Mathématique et d'Informatique, Avenue Maistriau, 15, 7000 Mons, Belgium
  2. Tbilisi State University, Chavchavadze 1, 380028 Tbilisi, Republic of Georgia

Abstract

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.

Keywords

ENG
Fourier, Fourier-Walsh series, Paley function, Haar system, separable Orlicz space

Bibliography

  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).
  2. Bloom S., and Kerman R., Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal function, Studia Math. 110 (1994), 149-167.
  3. Burkholder D., Distribution function inequalities for martingales, Ann. Probab. 11 (1973), 19-42.
  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.
  5. Ciesielski Z., and Kwapień S., Some properties of the Haar, Walsh-Paley, Franklin and the bounded polygonal orthonormal bases in Lp spaces, Comment. Math., Tomus Specialis in Honorem L. Orlicz, 1979, part 2, 37-42.
  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.
  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.
  8. Kashin B. S., and Saakjan A. A., Orthogonal Series, Amer. Math. Soc., Providence, 1989.
  9. Kolmogoroff A. N., Sur les fonctions harmoniques conjuguées et les séries de Fourier, Fund. Math. 7 (1927), 25-28.
  10. Komissarov A. A., Equivalence of Haar and Franklin systems in some function spaces, Sibirsk. Mat. Zh. 23 (5) (1982), 115-126 (in Russian).
  11. Konyagin S. V., On subsequences of partial Fourier-Walsh sums, Mat. Zametki 54 (4) (1993), 69-75 (in Russian).
  12. Krasnosel'skiĭ M. A., and Rutickiĭ Ya. B., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
  13. Lozinski S., On convergence and summability of Fourier series and interpolation processes, Mat. Sb. 14 (3) (1944), 175-268.
  14. Paley R. E. A. C., A remarkable system of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241-279.
  15. Riesz M., Sur les fonctions conjuguées, Math. Z. 27 (1927), 218-244.
  16. Ryan R., Conjugate functions in Orlicz spaces, Pacific J. Math. 13 (1963), 1371-1377.
  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.
  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).
  19. Watari C., Mean convergence of Walsh-Fourier series, Tôhoku Math. J. 16 (1964), 183-188.
  20. Zygmund A., Trigonometric Series, Vols. I, II, Cambridge Univ. Press, 1968.
Studia Mathematica
1996/121/2
Export to PBN, Opens in new tab
Pages:
193-205
Main language of publication
English
Received
1996-04-04
Published
1996
Exact and natural sciences