Geodesic
Optimal reconstruction of a~Banach function space from a~cone of nonnegative functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 142-156

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We study the problem of constructing a minimal Banach function space containing a given cone of nonnegative measurable functions. For the associate function norm of the norm of an optimal space, we obtain general formulas and specify them in the case of a cone defined by an integral representation. We also consider the similar problem of constructing an optimal rearrangement invariant space and compare the descriptions obtained. 
@article{TM_2014_284_a7,
     author = {M. L. Goldman and P. P. Zabreiko},
     title = {Optimal reconstruction of {a~Banach} function space from a~cone of nonnegative functions},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {142--156},
     publisher = {mathdoc},
     volume = {284},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2014_284_a7/}
}
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M. L. Goldman; P. P. Zabreiko. Optimal reconstruction of a~Banach function space from a~cone of nonnegative functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 142-156. http://geodesic.mathdoc.fr/item/TM_2014_284_a7/