Self-similar functions, fractals and algebraic genericity

Cariello, D.; Fávaro, V. V.; Seoane-Sepúlveda, J. B.

doi:10.1090/proc/13552

Self-similar functions, fractals and algebraic genericity
HTML articles powered by AMS MathViewer

by D. Cariello, V. V. Fávaro and J. B. Seoane-Sepúlveda PDF

Proc. Amer. Math. Soc. 145 (2017), 4151-4159 Request permission

Abstract:

We introduce the class of everywhere like functions, which helps us to recover some known classes (such as that of everywhere surjective ones). We also study the algebraic genericity of this new class together with the class of fractal functions.

References

N. Albuquerque, L. Bernal-González, D. Pellegrino, and J. B. Seoane-Sepúlveda, Peano curves on topological vector spaces, Linear Algebra Appl. 460 (2014), 81–96. MR 3250532, DOI 10.1016/j.laa.2014.07.029
Richard M. Aron, Luis Bernal González, Daniel M. Pellegrino, and Juan B. Seoane Sepúlveda, Lineability: the search for linearity in mathematics, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. MR 3445906
Richard M. Aron, José A. Conejero, Alfredo Peris, and Juan B. Seoane-Sepúlveda, Uncountably generated algebras of everywhere surjective functions, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 3, 571–575. MR 2731374
Richard Aron, V. I. Gurariy, and J. B. Seoane, Lineability and spaceability of sets of functions on $\Bbb R$, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795–803. MR 2113929, DOI 10.1090/S0002-9939-04-07533-1
Richard M. Aron and Juan B. Seoane-Sepúlveda, Algebrability of the set of everywhere surjective functions on $\Bbb C$, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 25–31. MR 2327324
F. Bastin, J. A. Conejero, C. Esser, and J. B. Seoane-Sepúlveda, Algebrability and nowhere Gevrey differentiability, Israel J. Math. 205 (2015), no. 1, 127–143. MR 3314585, DOI 10.1007/s11856-014-1104-1
Luis Bernal-González and Manuel Ordóñez Cabrera, Lineability criteria, with applications, J. Funct. Anal. 266 (2014), no. 6, 3997–4025. MR 3165251, DOI 10.1016/j.jfa.2013.11.014
Luis Bernal-González, Daniel Pellegrino, and Juan B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 1, 71–130. MR 3119823, DOI 10.1090/S0273-0979-2013-01421-6
F. J. Bertoloto, G. Botelho, V. V. Fávaro, and A. M. Jatobá, Hypercyclicity of convolution operators on spaces of entire functions, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 4, 1263–1283 (English, with English and French summaries). MR 3137355, DOI 10.5802/aif.2803
Daniel Cariello and Juan B. Seoane-Sepúlveda, Basic sequences and spaceability in $\ell _p$ spaces, J. Funct. Anal. 266 (2014), no. 6, 3797–3814. MR 3165243, DOI 10.1016/j.jfa.2013.12.011
Krzysztof Chris Ciesielski, José L. Gámez-Merino, Daniel Pellegrino, and Juan B. Seoane-Sepúlveda, Lineability, spaceability, and additivity cardinals for Darboux-like functions, Linear Algebra Appl. 440 (2014), 307–317. MR 3134273, DOI 10.1016/j.laa.2013.10.033
K. C. Ciesielski, J. L. Gámez-Merino, L. Mazza, and J. B. Seoane-Sepúlveda, Cardinal coefficients related to surjectivity, Darboux, and Sierpiński-Zygmund maps, Proc. Amer. Math. Soc. 145 (2017), no. 3, 1041–1052. MR 3589304, DOI 10.1090/proc/13294
J.A. Conejero, J.B. Seoane-Sepúlveda, and P. Sevilla-Peris, Isomorphic copies of $\ell _1$ for $m$-homogeneous non-analytic Bohnenblust–Hille polynomials, Math. Nachr. (in press).
Per H. Enflo, Vladimir I. Gurariy, and Juan B. Seoane-Sepúlveda, Some results and open questions on spaceability in function spaces, Trans. Amer. Math. Soc. 366 (2014), no. 2, 611–625. MR 3130310, DOI 10.1090/S0002-9947-2013-05747-9
Vladimir P. Fonf and Clemente Zanco, Almost overcomplete and almost overtotal sequences in Banach spaces, J. Math. Anal. Appl. 420 (2014), no. 1, 94–101. MR 3229811, DOI 10.1016/j.jmaa.2014.05.045
José L. Gámez-Merino, Gustavo A. Muñoz-Fernández, Víctor M. Sánchez, and Juan B. Seoane-Sepúlveda, Sierpiński-Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3863–3876. MR 2679609, DOI 10.1090/S0002-9939-2010-10420-3
José L. Gámez, Gustavo A. Muñoz-Fernández, and Juan B. Seoane-Sepúlveda, Lineability and additivity in $\Bbb R^{\Bbb R}$, J. Math. Anal. Appl. 369 (2010), no. 1, 265–272. MR 2643865, DOI 10.1016/j.jmaa.2010.03.036
José L. Gámez-Merino and Juan B. Seoane-Sepúlveda, An undecidable case of lineability in $\Bbb {R}^{\Bbb {R}}$, J. Math. Anal. Appl. 401 (2013), no. 2, 959–962. MR 3018041, DOI 10.1016/j.jmaa.2012.10.067
Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in analysis, Dover Publications, Inc., Mineola, NY, 2003. Corrected reprint of the second (1965) edition. MR 1996162
V. I. Gurariĭ, Linear spaces composed of everywhere nondifferentiable functions, C. R. Acad. Bulgare Sci. 44 (1991), no. 5, 13–16 (Russian). MR 1127022
F. B. Jones, Connected and disconnected plane sets and the functional equation $f(x)+f(y)=f(x+y)$, Bull. Amer. Math. Soc. 48 (1942), 115–120. MR 5906, DOI 10.1090/S0002-9904-1942-07615-4
Péter Komjáth and Vilmos Totik, Problems and theorems in classical set theory, Problem Books in Mathematics, Springer, New York, 2006. MR 2220838
Henri Leon Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives professées au Collège de France, Cambridge Library Collection, Cambridge University Press, Cambridge, 2009 (French). Reprint of the 1904 original. MR 2857993, DOI 10.1017/CBO9780511701825
J. Lukeš, G. A. Muñoz-Fernández, P. Petráček, and J. B. Seoane-Sepúlveda, Locally recurrent functions, density topologies and algebraic genericity, J. Math. Anal. Appl. 443 (2016), no. 2, 1247–1259. MR 3514345, DOI 10.1016/j.jmaa.2016.06.009
Juan B. Seoane, Chaos and lineability of pathological phenomena in analysis, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–Kent State University. MR 2709064