In this paper we introduce the notion of $\Phi$-bounded variation for metric space valued mappings defined on a subset of the real line. Such a notion generalizes the one for real functions introduced by M. Schramm, and many previous generalized variations. We prove a structural theorem for mappings of $\Phi$-bounded variation. As an application we show that each mapping of $\Phi$-bounded variation defined on a subset of $\RB$ possesses a $\Phi$-variation preserving extension to the whole real line.
Maniscalco, C. (2009). A Structural Theorem for Metric Space Valued Mappings of Φ-bounded Variation. REAL ANALYSIS EXCHANGE, 35(1), 79-90.
A Structural Theorem for Metric Space Valued Mappings of Φ-bounded Variation
MANISCALCO, Caterina
2009-01-01
Abstract
In this paper we introduce the notion of $\Phi$-bounded variation for metric space valued mappings defined on a subset of the real line. Such a notion generalizes the one for real functions introduced by M. Schramm, and many previous generalized variations. We prove a structural theorem for mappings of $\Phi$-bounded variation. As an application we show that each mapping of $\Phi$-bounded variation defined on a subset of $\RB$ possesses a $\Phi$-variation preserving extension to the whole real line.
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