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- W2086661903 abstract "Let (Ω, A , μ) be a finite measure space and let Φ : [0, ∞)→[0, ∞) be a Δ 2 -convex function, Φ≢0, Φ(0)=0. The Φ-approximation of a real A -measurable function f is the process of minimizing ∫ Ω Φ(| f − h |) d μ among the real functions h of some class M . In this paper we consider a class M of Σ-measurable functions, where Σ⊂ A is a σ-lattice that is totally ordered by a closed subset of the interval [0, 1]. A special example of M is the set of radial—nondecreasing or nonincreasing—functions defined on the unit ball of R n , and where μ is the Lebesgue measure. The primary purpose of this paper is to show a somewhat strong relation between Φ-approximation by M and Φ-approximation by constants on subsets of Ω. This assertion is enlightened by the following facts. Best step monotone Φ-approximants are formed by best constant Φ-approximants on subsets of Ω. Furthermore, appropriate sequences of best step monotone Φ-approximants have subsequences that converge a.e. to a best monotone Φ-approximant, and conversely, any best monotone Φ-approximant is a limit of a sequence of best step monotone Φ-approximants. On the one hand, these facts are applied to give methods for constructing best Φ-approximants. On the other hand, they yield rather theoretical results when M is, more generally, the class of Σ′-measurable functions, where Σ′⊂ A is an arbitrary σ-lattice." @default.
- W2086661903 created "2016-06-24" @default.
- W2086661903 creator A5074322106 @default.
- W2086661903 date "1996-04-01" @default.
- W2086661903 modified "2023-09-28" @default.
- W2086661903 title "Structure of Best Radial Monotone Φ-Approximants" @default.
- W2086661903 doi "https://doi.org/10.1006/jmaa.1996.0159" @default.
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