Composing functions of bounded $\varphi$-variation
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- by J. Ciemnoczołowski and W. Orlicz PDF
- Proc. Amer. Math. Soc. 96 (1986), 431-436 Request permission
Abstract:
Let ${F_n}$ be finite-valued functions on $( - \infty ,\infty ),{\text { }}{F_n}(0) = 0,{\text { }}n = 1,2, \ldots .$. For $x \in {\mathcal {V}_\varphi }\left \langle {a,b} \right \rangle$, the class of functions of bounded $\varphi$-variation, the compositions ${F_n}(x)$ are studied. The main result of this paper is Theorem 1 stating necessary and sufficient conditions for the sequence ${\operatorname {var} _\psi }({F_n}(x),a,b)$ to be bounded for each $x \in {\mathcal {V}_\varphi }\left \langle {a,b} \right \rangle$ ($\psi$ denotes here another $\varphi$-function).References
- Michael Josephy, Composing functions of bounded variation, Proc. Amer. Math. Soc. 83 (1981), no. 2, 354–356. MR 624930, DOI 10.1090/S0002-9939-1981-0624930-9
- R. Lésniewicz and W. Orlicz, On generalized variations. II, Studia Math. 45 (1973), 71–109. MR 346509, DOI 10.4064/sm-45-1-71-109 J. Marcinkiewicz, On a class of functions and their Fourier series, Collected Papers, Polish Scientific Publishers, Warszawa, 1964, pp. 36-41.
- J. Musielak and W. Orlicz, On generalized variations. I, Studia Math. 18 (1959), 11–41. MR 104771, DOI 10.4064/sm-18-1-11-41
- J. Musielak and W. Orlicz, On modular spaces, Studia Math. 18 (1959), 49–65. MR 101487, DOI 10.4064/sm-18-1-49-65 N. Wiener, The quadratic variation of a function and its Fourier coefficients, Massachusetts J. Math. 3 (1924), 72-94. L. C. Young, Inequalities connected with bounded $p$-th power variation in the Wiener sense and with integrated Lipschitz conditions, Proc. London Math. Soc. (2) 43 (1937), 449-467. —, Sur une généralisation de la notion de variation de puissance pième bornée au sens de N. Wiener et sur la convergence des séries de Fourier, C.R. Acad. Sci. Paris Sér. A 204 (1937), 470-472.
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 96 (1986), 431-436
- MSC: Primary 26A45; Secondary 26A16
- DOI: https://doi.org/10.1090/S0002-9939-1986-0822434-6
- MathSciNet review: 822434