Zipper Fractal Functions with Variable Scalings

Year 2022, , 481 - 501, 30.12.2022

. Vijay A. K. B. Chand

https://doi.org/10.31197/atnaa.1149689

Cited By: 4

Abstract

Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an
improved version of iterated function system by using a binary parameter called a signature. The signature
allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can
be non-differentiable on a dense subset of an interval I. In this paper, we construct k-times continuously
differentiable ZFIFs with variable scaling functions on I. Some properties like the positivity, monotonicity,
and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a
zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space
of k-times continuously differentiable functions and the space of p-integrable functions for p ∈ [1,∞) are
studied. We introduce the zipper versions of full Müntz theorem for continuous function and p-integrable
functions on I for p ∈ [1,∞).

Keywords

Fractals, zipper smooth fractal function, topological isomorphism, Schauder basis, linear operator

References

• [1] V.V. Aseev, On the regularity of self-similar zippers, 6th Russian-Korean International Symposium on Science and Tech- nology, KORUS-2002 (June 24-30, 2002, Novosibirsk State Techn. Univ. Russia, NGTU, Novosibirsk), Part 3 (Abstracts), 2002, p.167.
• [2] V.V. Aseev, A.V. Tetenov, and A.S. Kravchenko, On self-similar Jordan curves on the plane, Sib. Math. J. 44(3) (2003), 379–386.
• [3] M.F. Barnsley, Fractal functions and interpolation, Constr. Approx., 2(1) (1986), 303–329.
• [4] M.F. Barnsley and A.N. Harrington, The calculus of fractal interpolation functions, J. Approx. Theory, 57(1) (1989), 14–34.
• [5] S.N. Bernstein, Sur les recherches recentes relatives à la meilleure approximation des fonctions continues par les polynômes, in Proc. of 5th Inter. Math. Congress, 1 (1912), 256–266.
• [6] P. Borwein and T. Erdelyi, Polynomials and polynomial inequalities, Springer-Verlag, New York, 1996.
• [7] P. Borwein and T. Erdelyi, The Full Müntz Theorem in C[0,1] and L 1 (0,1), J. London Math. Soc., 54 (1996), 102–110.
• [8] A.K.B. Chand and G.P. Kapoor, Generalized cubic spline fractal interpolation functions, SIAM J. Numer. Anal., 44(2) (2006), 655–676.
• [9] A.K.B. Chand, N. Vijender, P. Viswanathan, and A .V Tetenov, Affine zipper fractal interpolation functions, BIT Num. Math., 60 (2020), 319–344.
• [10] E.W. Cheney, Approximation theory, AMS Chelsea Publishing Company, Providence, RI, 1966.
• [11] P.J. Davis, Interpolation and Approximation, Dover, 1975.
• [12] C. Heil, A basis theory primer, Birkhäuser, Bostan, 2011.
• [13] J.E. Hutchinson, Fractals and self-similarity, Indiana U. Math. J., 30(5) (1981), 713–747.
• [14] S. Jha, A. K. B. Chand, and M. A. Navascu´ es, Approximation by shape preserving fractal functions with variable scaling, Calcolo, 58(8) (2021). https://doi.org/10.1007/s10092-021-00396-8
• [15] B. Mandelbrot, Fractals: Form, Chance and Dimension, W. H. Freeman, San Francisco, 1977.
• [16] Ch. H. Müntz, Uber den approximationssatz von Weierstrass, in H. A. Schwarz’s Festschrift, Berlin, (1914), 303–312.
• [17] M.A. Navascues, Fractal polynomial interpolation, Z. Anal. Anwend, 24(2) (2005), 1–20.
• [18] M.A. Navascues, Fractal approximation, Complex Anal. Oper. Theory, 4(4) (2010), 953–974.
• [19] M.A. Navascues, Fractal bases for L p spaces, World Scientific Publishing Company, 20(2) (2012), 141–148.
• [20] M.A. Navascues, Affine fractal functions as bases of continuous functions, Quaestiones Mathematicae, 37(3) (2014), 415–428.
• [21] M. A. Navascues and A. K. B. Chand, Fundamental sets of fractal functions, Acta Appl. Math., 100 (2018), 247–261.
• [22] M.A. Navascues, P. Viswanathan, A. K. B. Chand, M. V. Sebastián, and S. K. Katiyar, Fractal bases for banach spaces of smooth functions, Bull. Aust. Math. Soc., 92(3) (2015), 405–419.
• [23] V. Operstein, Full Müntz Theorem in L p (0,1), J. Approx. Theory, 85 (1996), 233–235.
• [24] K. M. Reddy, Some aspects of fractal functions in geometric modelling, Ph.D. Thesis, IIT Madras, 2018.
• [25] S. Schonefeld, Schauder basis in spaces of differentiable functions, Bull. Amer. Math. Soc., 75 (1969), 586–590.
• [26] L. Schwartz, Etude des sommes D’Exponentielles, Hermann, Paris, 1959.
• [27] A.R. Siegel, On the Müntz-Szász Theorem for C[0,1], Proc. Amer. Math. Soc., 36 (1972).
• [28] O. Szász, Uber die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Math. Ann., 77 (1916), 482–496.
• [29] O. Szász, On closed sets of rational functions, Ann. Mat. Pura. Appl., 34 (1953), 195–218.
• [30] A.V. Tetenov, On self-similar Jordan arcs on a plane, Sib. Zh. Ind. Mat., 7(3) (2004), 148–155.
• [31] A.V. Tetenov, Self-similar Jordan arcs and graph-directed systems of similarities, Siberian Math. J., 47(5) (2006), 940–949.
• [32] A.V. Tetenov, M. Samuel and D. A. Vaulin, Self-similar dendrites generated by polygonal systems in the plane, Sib. Elektron. Mat. Izv., 14 (2017), 737–751.
• [33] P. Viswanathan, M.A. Navascu´ es, and A.K.B. Chand, Associate fractal functions in Lp -spaces and in one-sided uniform approximation, Journal of Mathematical Analysis and Applications, 433(2) (2016), 862–876.
• [34] H.Y. Wang and J.S. Yu, Fractal interpolation functions with variable parameters and their analytical properties, J. Approx. Theory, 175 (2013), 1–18.
• [35] K. Yosida, Functional analysis, Berlin Heidelberg, New York, 1980.