Sigma order continuity and best approximation in L ϱ -spaces

Shelby J. Kilmer; Wojciech M. Kozƚowski; Grzegorz Lewicki

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 2, page 241-250
  • ISSN: 0010-2628

Abstract

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In this paper we give a characterization of σ -order continuity of modular function spaces L ϱ in terms of the existence of best approximants by elements of order closed sublattices of L ϱ . We consider separately the case of Musielak–Orlicz spaces generated by non- σ -finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.

How to cite

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Kilmer, Shelby J., Kozƚowski, Wojciech M., and Lewicki, Grzegorz. "Sigma order continuity and best approximation in $L_\varrho $-spaces." Commentationes Mathematicae Universitatis Carolinae 32.2 (1991): 241-250. <http://eudml.org/doc/247255>.

@article{Kilmer1991,
abstract = {In this paper we give a characterization of $\sigma $-order continuity of modular function spaces $L_\varrho $ in terms of the existence of best approximants by elements of order closed sublattices of $L_\varrho \,$. We consider separately the case of Musielak–Orlicz spaces generated by non-$\sigma $-finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.},
author = {Kilmer, Shelby J., Kozƚowski, Wojciech M., Lewicki, Grzegorz},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {best approximation; lattices; modular function spaces; $L_\varrho $-spaces; Orlicz spaces; -order continuity; modular function space; closed sublattices; Musielak-Orlicz spaces},
language = {eng},
number = {2},
pages = {241-250},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Sigma order continuity and best approximation in $L_\varrho $-spaces},
url = {http://eudml.org/doc/247255},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Kilmer, Shelby J.
AU - Kozƚowski, Wojciech M.
AU - Lewicki, Grzegorz
TI - Sigma order continuity and best approximation in $L_\varrho $-spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 2
SP - 241
EP - 250
AB - In this paper we give a characterization of $\sigma $-order continuity of modular function spaces $L_\varrho $ in terms of the existence of best approximants by elements of order closed sublattices of $L_\varrho \,$. We consider separately the case of Musielak–Orlicz spaces generated by non-$\sigma $-finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.
LA - eng
KW - best approximation; lattices; modular function spaces; $L_\varrho $-spaces; Orlicz spaces; -order continuity; modular function space; closed sublattices; Musielak-Orlicz spaces
UR - http://eudml.org/doc/247255
ER -

References

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  1. Ando T., Amemiya I., Almost everywhere convergence of prediction sequences in L p ( 1 < p < ∞ ) , Z. Wahrscheinlichkeitstheorie verw. Gebiete 4 (1965), 113-120. (1965) MR0189077
  2. Birnbaum Z., Orlicz W., Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen, Studia Math. 3 (1931), 1-67. (1931) Zbl0003.25202
  3. Kilmer S., Kozlowski W.M., Lewicki G., Best approximants in modular function spaces, Journ. Approx. Th. 63 (1990), 338-367. (1990) Zbl0718.41049MR1081035
  4. Kozlowski W.M., Modular Function Spaces, Series of Monographs and Textbooks in Pure and Applied Mathematics, 122, Dekker, New York, Basel, 1988. Zbl0718.41049MR1474499
  5. Kozlowski W.M., Notes on modular function spaces I, Comment. Math. 28 (1988), 87-100. (1988) Zbl0747.46023MR0988963
  6. Kozlowski W.M., Notes on modular function spaces II, Comment. Math. 28 (1988), 101-116. (1988) Zbl0747.46022MR1474499
  7. Krasnosel'skii M.A., Rutickii Y.B., Convex functions and Orlicz spaces, Noordhoff, Groningen, 1961. MR0126722
  8. Landers D., Rogge L., Best approximants in L ϕ -spaces, Z. Wahrscheinlichkeitstheorie verw. Gebiete 51 (1980), 215-237. (1980) MR0566317
  9. Lindenstrauss J., Tzafriri J., Classical Banach Spaces II: Function Spaces, Springer-Verlag, Berlin, Heidelberg, New York, 1979. Zbl0403.46022MR0540367
  10. Luxemburg W., Zaanen A., Notes on Banach function spaces I-XIII, Proc. Royal Acad. Sci., Amsterdam (1963) A-66, 135-153, 239-263, 496-504, 655-681; (1964) A-64, 101-119; (1964) A-67, 360-376, 493-543. MR0173167
  11. Musielak J., Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. Zbl0557.46020MR0724434
  12. Orlicz W., Über eine gewisse Klasse von Räumen vom Typus B, Bull. Acad. Polon. Sci. Ser A (1932), 207-220. (1932) Zbl0006.31503
  13. Orlicz W., Über Räumen L M , Bull. Acad. Polon. Sci. Ser A (1936), 93-107. (1936) 
  14. Shintani T., Ando T., Best approximants in L 1 space, Z. Wahrscheinlichkeitstheorie verw. Gebiete 33 (1975), 33-39. (1975) Zbl0298.41016MR0380963
  15. Turett B., Fenchel-Orlicz spaces, Dissertationes Math. 181 (1980), 1-60. (1980) Zbl0435.46025MR0578390
  16. Zaanen A.C., Integration, North Holland, Amsterdam, London, 1967. Zbl0671.42001MR0222234

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