Abstract
Given a continuous function f defined on the unit cube ofR n and a convex function φt, φt(0)=0, φt(x)>0, for x>0, we prove that the set of best Lφt-approximations by monotone functions has exactly one element ft, which is also a continuous function. Moreover if the family of convex functions {φt}t>0 converges uniformly on compact sets to a function φ0, then the best approximation ft→f0 uniformly, as t→0, where f0 is the best approximation of f within the Orlicz space L⃜0. The best approximations {ft} are obtained as well as minimizing integrals or the Luxemburg norm.
Similar content being viewed by others
References
Darst, R.B. and Huotari, R., MonotoneL 1-Approximation on The Unitn-Cube, Proc. Amer. Math. Soc. 95:3 (1985), 425–428.
Huotari, R. and Legg, D., Monotone Approximation in Several Variables, J. of Approx. Theory. 47:3(1986), 219–227.
Krasnoselskii, M. A. and Rutickii, Ya. B., Convex functions and Orlicz spaces, P. Noordhoff, Groningen, 1961.
Landers, D. and Rogge, L., Best Approximants inL ⃜-Spaces, Z. Wahrsch., Verw. Gebiete., 51(1980), 215–237.
Musielak, J., Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics 1034, Springer Verlag. Berlin-Heidelberg-N. Y., 1983.
Rao, M. M. and Ren, Z. D., Theory of Orlicz Spaces, Pure and Applied Mathematics, Marcel Dekker, Inc., 1991.
Fernández, F. Zó, C. and Favier, S., The Natural ║ ║⃜-approximant in Orlicz spaces, Rev. Unión Mat. Arg., 39 (1994), 27–44.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Iturrieta, M., Zó, F. Best monotoneL φ-approximations in several variables. Approx. Theory & its Appl. 14, 1–10 (1998). https://doi.org/10.1007/BF02836763
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02836763