A modulus of continuity for a class of quasismooth functions
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- by J. Ernest Wilkins PDF
- Proc. Amer. Math. Soc. 93 (1985), 459-465 Request permission
Abstract:
Let $Z$ be the class of real-valued functions $f(x)$, defined and continuous on the closed interval $I = [ - 1,1]$, such that $f( - 1) = f(1) = 0$ and \[ |f(\xi ) - 2f\{ (\xi + \eta )/2\} + f(\eta )| \leqslant |\xi - \eta |\] for all $\xi$ and $\eta$ in $I$. We show that $\omega (h) = h{\log _2}\left \{ {2eK/(h{{\log }_2}e)} \right \}$ is a modulus of continuity on $Z$, if $K = {\sup _{f \in Z}}{\max _{x \in I}}\left | {f(x)} \right |$.References
- Yu. A. Brudnyĭ, On the maximum modulus of a quasi-smooth function, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 4(76), 273–275 (Russian). MR 0089876
- I. P. Sokolova, The maximum of the modulus of a function that satisfies Zygmund’s condition, Studies in the theory of functions of several real variables (Russian), Jaroslav. Gos. Univ., Yaroslavl, 1976, pp. 66–71 (Russian). MR 0580933
- A. F. Timan, Quasi-smooth functions, Doklady Akad. Nauk SSSR (N.S.) 70 (1950), 961–963 (Russian). MR 0033321
- A. F. Timan, Quasi-smooth functions, Uspehi Matem. Nauk (N.S.) 5 (1950), no. 3(37), 128–130 (Russian). MR 0035801
- A. F. Timan, On quasi-smooth functions, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 243–254 (Russian). MR 0041899
- J. Ernest Wilkins Jr. and Theodore R. Hatcher, The maximum of a quasismooth function, Math. Comp. 41 (1983), no. 164, 573–589. MR 717704, DOI 10.1090/S0025-5718-1983-0717704-1
- A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47–76. MR 12691
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 459-465
- MSC: Primary 26A15; Secondary 26D20
- DOI: https://doi.org/10.1090/S0002-9939-1985-0774003-3
- MathSciNet review: 774003