Abstract

Let ff be a continuous real function on a convex subset of a Banach space. We study what can be said about the semiconcavity (with a general modulus) of ff, if we know that the estimate Δh2(f,x)≤ω(∥h∥)\Delta_h^2(f,x) \leq \omega(\|h\|) holds, where Δh2(f,x)=f(x+2h)−2f(x+h)+f(x)\Delta_h^2(f,x) = f(x+2h)-2f(x+h) + f(x) and ω:[0,∞)→[0,∞)\omega:[0,\infty) \to [0,\infty) is a nondecreasing function right continuous at 00 with ω(0)=0\omega(0) =0. A partial answer to this question was given by P. Cannarsa and C. Sinestrari (2004); we prove versions of their result, which are in a sense best possible. We essentially use methods of A. Marchaud, S. B. Stechkin and others, whose results clarify when the inequality ∣Δh2(f,x)∣≤ω(∥h∥)|\Delta_h^2(f,x)| \leq \omega(\|h\|) implies that ff is a C1C^1 function (and f′f' is uniformly continuous with a corresponding modulus of continuity).

Author information

Contact details are reproduced from the original publication and may be historical.

Luděk Zajíček

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

zajicek@karlin.mff.cuni.cz

Keywords

  • Semiconcave function with general modulus
  • second difference
  • second modulus of continuity
  • Jensen semiconcave function
  • alpha-midconvex function
  • semi-Zygmund class

Mathematics Subject Classification

26B25; 46T99

References

  1. N. K. Bari, S. B. Stechkin: Best approximations and differential properties of two conjugate functions (Russian), Trudy Moskov. Mat. Obs. 5 (1956) 483–522.Google Scholar
  2. Ch. S. Allen: A characterization of dimension, Amer. Math. Monthly 93 (1986) 635–636.https://doi.org/10.1080/00029890.1986.11971905Google Scholar
  3. P. Cannarsa, C. Sinestrari: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications 58, Birkhäuser, Boston (2004).https://doi.org/10.1007/b138356Google Scholar
  4. V. K. Dzyadyk, I. A. Shevchuk: Theory of Uniform Approximation of Functions by Polynomials, Walter de Gruyter, Berlin (2008).https://doi.org/10.1515/9783110208245Google Scholar
  5. R. A. DeVore, G. G. Lorentz: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften 303, Springer, Berlin (1993).https://doi.org/10.1007/978-3-662-02888-9_10Google Scholar
  6. J. Duda, L. Zajíček: Semiconvex functions: representations as suprema of smooth functions and extensions, J. Convex Analysis 16 (2009) 239–260.https://doi.org/10.68381/jca16013Google Scholar
  7. J. Duda, L. Zajíček: Smallness of singular sets of semiconvex functions in separable Banach spaces, J. Convex Analysis 20 (2013) 573–598.https://doi.org/10.68381/jca20034Google Scholar
  8. L. G. Hanin: Closed ideals in algebras of smooth functions, Dissertationes Math. (Rozprawy Mat.) 371 (1998) 67 pp.Google Scholar
  9. M. Johanis, L. Zajíček: Smoothness via directional smoothness and Marchaud’s theorem in Banach spaces, J. Math. Anal. Appl. 423 (2015) 594–607.https://doi.org/10.1016/j.jmaa.2014.09.068Google Scholar
  10. A. Marchaud: Sur les dérivées et sur les différences des fonctions de variables réelles, J. Math. Pures. Appl., Sér. 9, 6 (1927) 337–425.Google Scholar
  11. S. B. Stechkin: On the order of the best approximations of continuous functions (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 15 (1951) 219–242.Google Scholar
  12. S. B. Stechkin: On absolute convergence of Fourier series II (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 19 (1955) 221–246.Google Scholar
  13. J. Tabor, J. Tabor: Generalized approximate midconvexity, Control Cybernet. 38 (2009) 655–669.Google Scholar
  14. J. Tabor, J. Tabor, A. Murenko: Semiconcave functions with power moduli, J. Convex Analysis 18 (2011) 391–396.https://doi.org/10.68381/jca18022Google Scholar
  15. L. Zajíček: Differentiability of approximately convex, semiconcave and strongly paraconvex functions, J. Convex Analysis 15 (2008) 1–15.https://doi.org/10.68381/jca15001Google Scholar

Suggested citation

L. Zajíček. “On Semiconcavity via the Second Difference.” Journal of Convex Analysis 25 (2018), No. 1, 241–269. https://doi.org/10.68381/jca25015

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press LLC.