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Keywords:
best approximation; lattices; modular function spaces; $L_\varrho $-spaces; Orlicz spaces
best approximation; lattices; modular function spaces; $L_\varrho $-spaces; Orlicz spaces
Summary:
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.
References:
[1] Ando T., Amemiya I.: Almost everywhere convergence of prediction sequences in $L_p(1< p<\infty)$. Z. Wahrscheinlichkeitstheorie verw. Gebiete 4 (1965), 113-120. MR 0189077
[2] Birnbaum Z., Orlicz W.: Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen. Studia Math. 3 (1931), 1-67. Zbl 0003.25202
[3] Kilmer S., Kozlowski W.M., Lewicki G.: Best approximants in modular function spaces. Journ. Approx. Th. 63 (1990), 338-367. MR 1081035 | Zbl 0718.41049
[4] Kozlowski W.M.: Modular Function Spaces. Series of Monographs and Textbooks in Pure and Applied Mathematics, 122, Dekker, New York, Basel, 1988. MR 1474499 | Zbl 0718.41049
[5] Kozlowski W.M.: Notes on modular function spaces I. Comment. Math. 28 (1988), 87-100. MR 0988963 | Zbl 0747.46023
[6] Kozlowski W.M.: Notes on modular function spaces II. Comment. Math. 28 (1988), 101-116. MR 1474499 | Zbl 0747.46022
[7] Krasnosel'skii M.A., Rutickii Y.B.: Convex functions and Orlicz spaces. Noordhoff, Groningen, 1961. MR 0126722
[8] Landers D., Rogge L.: Best approximants in $L_\varphi $-spaces. Z. Wahrscheinlichkeitstheorie verw. Gebiete 51 (1980), 215-237. MR 0566317
[9] Lindenstrauss J., Tzafriri J.: Classical Banach Spaces II: Function Spaces. Springer-Verlag, Berlin, Heidelberg, New York, 1979. MR 0540367 | Zbl 0403.46022
[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. MR 0173167
[11] Musielak J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. MR 0724434 | Zbl 0557.46020
[12] Orlicz W.: Über eine gewisse Klasse von Räumen vom Typus B. Bull. Acad. Polon. Sci. Ser A (1932), 207-220. Zbl 0006.31503
[13] Orlicz W.: Über Räumen $L^M$. Bull. Acad. Polon. Sci. Ser A (1936), 93-107.
[14] Shintani T., Ando T.: Best approximants in $L_1$ space. Z. Wahrscheinlichkeitstheorie verw. Gebiete 33 (1975), 33-39. MR 0380963 | Zbl 0298.41016
[15] Turett B.: Fenchel-Orlicz spaces. Dissertationes Math. 181 (1980), 1-60. MR 0578390 | Zbl 0435.46025
[16] Zaanen A.C.: Integration. North Holland, Amsterdam, London, 1967. MR 0222234 | Zbl 0671.42001
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