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Characterization of the \(\text{RV}_{\phi, 2, \alpha}([a, b])\) space. (English) Zbl 1452.26011

Summary: We introduce the \((\phi, 2, \alpha)\)-bounded variation of a function, a concept that generalizes the \(\phi\)-bounded variation introduced by Medvedev. We also provide a characterization to find out the \((\phi, 2, \alpha)\)-variation of a function \(f\) in the sense of Riesz.

MSC:

26A45 Functions of bounded variation, generalizations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

[1] J. Appell, J. Banaś, and N. Merentes, Bounded variation and around, De Gruyter Series in Nonlinear Analysis and Applications 17, De Gruyter, 2014. Mathematical Reviews (MathSciNet): MR3156940
· Zbl 1282.26001
[2] R. E. Castillo and E. Trousselot, “On functions of \((p,\alpha)\)-bounded variation”, Real Anal. Exchange 34:1 (2009), 49-60. Mathematical Reviews (MathSciNet): MR2527121
Zentralblatt MATH: 1179.26032
Digital Object Identifier: doi:10.14321/realanalexch.34.1.0049
Project Euclid: euclid.rae/1242738919
· Zbl 1179.26032 · doi:10.14321/realanalexch.34.1.0049
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Zentralblatt MATH: 1298.26032
· Zbl 1298.26032
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Zentralblatt MATH: 1343.46033
· Zbl 1343.46033
[5] R. E. Castillo, H. C. Chaparro, and E. Trousselot, “On functions of \((\phi, 2, \alpha)\)-bounded variation”, submitted, 2020. Mathematical Reviews (MathSciNet): MR2527121
Digital Object Identifier: doi:10.14321/realanalexch.34.1.0049
Project Euclid: euclid.rae/1242738919
· Zbl 1179.26032 · doi:10.14321/realanalexch.34.1.0049
[6] C. Jordan, “Sur la série de Fourier”, C. R. Acad. Sci. 92 (1881), 228-230. · JFM 13.0184.01
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[8] F. Riesz, “Untersuchungen über Systeme integrierbarer Funktionen”, Math. Ann. 69:4 (1910), 449-497. Mathematical Reviews (MathSciNet): MR1511596
Digital Object Identifier: doi:10.1007/BF01457637
· JFM 41.0383.01 · doi:10.1007/BF01457637
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