Paraconvex, but not strongly, Takagi functions

• Autorzy: Tabor J. , Tabor J.
• Języki publikacji: EN

Abstrakty

EN

There is an important open problem in the theory of approximate convexity whether every paraconvex function on a bounded interval is strongly paraconvex. Our aim is to show that this is not the case. To do this we need the following generalization of Takagi function. For a sequence a = (a_i)_i∈N ⊂ R₊ we consider Takagi-like function of the form T(a)(x) := ∑ ^∞_i=1 a_idist(x, ¹₂^i-1Z) for x ∈ R. We give convenient conditions for verification whether T(a) is paraconvex or strongly paraconvex. This enables us to construct a class of paraconvex functions which are not strongly paraconvex.

Słowa kluczowe

EN

paraconvexity; strongly paraconvex function; semiconcavity; Takagi function

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2012
• Tom: Vol. 41, no. 3
• Strony: 545--559
• Opis fizyczny: Bibliogr. 27 poz.

Twórcy

autor

Tabor J.

autor

Tabor J.

• Chair of Computational Mathematics, Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland,

Bibliografia

• Allaart P. C. and Kawamura K. (2010) The improper infinite derivatives of Takagi’s nowhere-differentiable function. J. Math. Anal. Appl. 372 (2) 656-665.
• Allaart P. C. and Kawamura K. (2011) The Takagi function: a survey. Real Anal. Exchange 37 (1) 1-54.
• Boros Z. (2008) An inequality for the Takagi function. Math. Inequal. Appl. 11, 751-765.
• Cannarsa P., Sinestrari C. (2004) Semiconcave Functions, Hamilton-Jacobi Equation and Optimal Control. Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser, Boston.
• Falconer K. (1990) Fractal Geometry. Mathematical Foundations and Applications. Wiley and Sons, Chichester.
• Hata M., Yamaguti M. (1984) The Takagi’s function and its generalization. Japan J. Appl. Math. 1, 183-199.
• Házy A. (2005) On approximate t-convexity. Mathematical Inequal. Appl. 8 (3) 389-402.
• Házy A., Páles Zs. (2004) On approximately midconvex functions. Bulletin London Math. Soc. 36 (3) 339-350.
• Kairies H.-H. (1997) Functional equations for peculiar functions. Aequationes Math. 53, 207-241.
• Kairies H.-H. (1998) Takagis function and its functional equations. Rocznik Nauk.-Dydakt. Prace Mat. 15, 73-83.
• Kôno N. (1987) On generalized Takagi functions. Acta Math. Hung. 49, 315-324.
• Krüppel M. (2007) On the extrema and the improper derivatives of Takagis continuous nowhere differentiable function. Rostocker Math. Kolloq. 62, 41-59.
• Krüppel M. (2008) Takagis continuous nowhere differentiable function and binary digital sums. Rostocker Math. Kolloq. 63, 37-54.
• Lagarias J. C. (2011) The Takagi’s function and its properties. arXiv.org no. 1112.4205.
• Makó J. and Páles Zs. (2010) Approximate convexity of Takagi type functions. J. Math. Anal. Appl. 369, 545-554.
• Ngai H., Luc D. T., Théra M. (2000) Approximately convex functions. Convex Anal. 1, 155-176.
• Páles Zs. (2003) On approximately convex functions. Proc. Amer. Math. Soc. 131 (1) 243-252.
• Rolewicz S. (1979) On γ-paraconvex multifunctions. Math. Japonica 24 (3) 293-300.
• Rolewicz S. (2000) On α(·)-paraconvex and strongly α(·)-paraconvex functions. Control and Cybernetics 29, 367-377.
• Rolewicz S. (2005) On differentiability of strongly α()-paraconvex functions in non-separable Asplund spaces. Studia Math. 167, 235-244.
• Rolewicz S. (2005) Paraconvex analysis. Control and Cybernetics 34, 951-965.
• Tabor Jacek and Tabor Józef (2009) Generalized approximate midconvexity. Control and Cybernetics 38 (3) 1-15.
• Tabor Jacek and Tabor Józef (2009) Takagi functions and approximate midconvexity. J. Math. Anal. Appl. 356 (2) 729-737.
• Takagi T. (1903) A simple example of continuous function without derivative. Proc. Phy.- Math. Soc. Japan 1, 176-177.
• Yamaguti M., Hata M., Kigami J. (1997) Mathematics of Fractals. AMS Translations of Mathematical Monographs 167.
• Zajíček L. (2007) A C1 function which is nowhere strongly paraconvex and nowhere semiconcave. Control and Cybernetics 36, 803-810.
• Zajíček L. (2008) Differentiability of approximately convex, semiconcave and strongly paraconvex functions. J. Convex Anal. 15 (1) 1-15.