Approximation by shape preserving fractal functions with variable scalings

Abstract

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions \(f^{\varvec{\alpha }}\in {\mathcal {C}}(I)\), where \(\varvec{\alpha }\) is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the \(\varvec{\alpha }\)-fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the \(\varvec{\alpha }\)-fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the \(\varvec{\alpha }\)-fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Müntz theorem in \({\mathcal {C}}[0,1]\).

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References

  1. 1.

    Ali, M., Clarkson, T.G.: Using linear fractal interpolation functions to compress video images. Fractals 2(03), 417–421 (1994)

    Article  Google Scholar 

  2. 2.

    Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2(4), 303–329 (1986)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Barnsley, M.F.: Fractals Everywhere. Academic Press Inc., Boston (1988)

    MATH  Google Scholar 

  4. 4.

    Barnsley, M.F., Vince, A.: Developments in fractal geometry. Bull. Math. Sci. 3(2), 299–348 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Basu, S., Foufoula-Georgiou, E., Porté-Agel, F.: Synthetic turbulence, fractal interpolation, and large-eddy simulation. Phys. Rev. E 70(2), 026310 (2004)

    Article  Google Scholar 

  6. 6.

    Borwein, P., Erdélyi, T.: The full Müntz theorem in \(C[0,1]\) and \(L_1[0,1]\). J. Lond. Math. Soc. (2) 54(1), 102–110 (1996)

    Article  Google Scholar 

  7. 7.

    Buescu, J., Serpa, C.: Fractal and hausdorff dimensions for systems of iterative functional equations. J. Math. Anal. Appl. 480, 76–83 (2019)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Carothers, N.L.: A short course on approximation theory. Bowling Green State University, Bowling Green (1998)

    Google Scholar 

  9. 9.

    Chand, A.K.B., Kapoor, G.P.: Generalized cubic spline fractal interpolation functions. SIAM J. Numer. Anal. 44(2), 655–676 (2006)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chand, A.K.B., Vijender, N., Navascués, M.A.: Shape preservation of scientific data through rational fractal splines. Calcolo 51(2), 329–362 (2014)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chand, A.K.B., Vijender, P., Viswanathan, N., Tetenov, A.V.: Affine zipper fractal interpolation functions. BIT Numer. Math. 60(2), 319–344 (2020)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Cheney, E.W.: Introduction to Approximation Theory. McGraw-Hill Book Co., New York (1966)

    MATH  Google Scholar 

  13. 13.

    Craciunescu, O.I., Das, S.K., Poulson, J.M., Samulski, T.V.: Three-dimensional tumor perfusion reconstruction using fractal interpolation functions. IEEE Trans. Biomed. Eng. 48(4), 462–473 (2001)

    Article  Google Scholar 

  14. 14.

    Dalla, L., Drakopoulos, V.: On the parameter identification problem in the plane and the polar fractal interpolation functions. J. Approx. Theory 101(2), 289–302 (1999)

    MathSciNet  Article  Google Scholar 

  15. 15.

    David, S.M., Moson, H.H.: Using iterated function systems to model discrete sequences. IEEE Trans. Signal Process. 40(7), 1724–1734 (1992)

    Article  Google Scholar 

  16. 16.

    Dudley, J.M., Finot, C., Richardson, D.J., Millot, G.: Self-similarity in ultrafast nonlinear optics. Nat. Phys. 3(9), 597–603 (2007)

    Article  Google Scholar 

  17. 17.

    Flint, G., Hambly, B., Lyons, T.: Discretely sampled signals and the rough Hoff process. Stoch. Process. Appl. 126(9), 2593–2614 (2016)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Ford, W.T., Roulier, J.A.: On interpolation and approximation by polynomials with monotone derivatives. J. Approx. Theory 10, 123–130 (1974)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hambly, B., Yang, W.: Existence and space–time regularity for stochastic heat equations on PCF fractals. Electron. J. Probab. 23(22), 33 (2018)

    Google Scholar 

  20. 20.

    Hu, Y., Leviatan, D., Yu, X.M.: Copositive polynomial approximation in \(C[0,1]\). J. Anal. 1, 85–90 (1993)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Leviatan, D.: Monotone and comonotone polynomial approximation revisited. J. Approx. Theory 53(1), 1–16 (1988)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Luor, D.C.: Fractal interpolation functions with partial self similarity. J. Math. Anal. Appl. 464(1), 911–923 (2018)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Masopust, P.R.: Fractal Functions, Fractal Surfaces, and Wavelets, 2nd edn. Elsevier/Academic Press, London (2016)

    Google Scholar 

  24. 24.

    Massopust, P.R.: Interpolation and Approximation with Splines and Fractals. Oxford University Press, Oxford (2010)

    MATH  Google Scholar 

  25. 25.

    Navascués, M.A.: Fractal polynomial interpolation. Z. Anal. Anwend. 24(2), 401–418 (2005)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Navascués, M.A.: Local variability of non-smooth functions. Nonlinear Anal. Theory Methods Appl. 70(7), 2506–2518 (2009)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Navascués, M.A., Jha, S., Chand, A.K.B., Sebastián, M.V.: Fractal approximation of Jackson type for periodic phenomena. Fractals 26(5), 1850079, 14 (2018)

  28. 28.

    Navascués, M.A., Sebastián, M.V.: Fitting curves by fractal interpolation: an application to the quantification of cognitive brain processes, pp. 143–154. World Scientific, Thinking In Patterns (2004)

  29. 29.

    Pál, J.: Approksimation af konvekse funktioner ved konvekse polynomier, Matematisk Tidsskrift. B 60–65 (1925)

  30. 30.

    Passow, E., Raymon, L., Shisha, O.: Piecewise monotone interpolation and approximation with Muntz polynomials. Trans. Am. Math. Soc. 218, 197–205 (1976)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Samuel, M., Tetenov, A.V.: On attractors of iterated function systems in uniform spaces. Sib. Èlektron. Mat. Izv. 14, 151–155 (2017)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Schuster, H.G., Just, W.: Deterministic Chaos: An Introduction. Wiley-VCH Verlag GmbH and Co, KGaA (2005)

    Book  Google Scholar 

  33. 33.

    Serpa, C., Buescu, J.: Explicitly defined fractal interpolation functions with variable parameters. Chaos Solitons Fractals 75, 76–83 (2015)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Simon, K.: The dimension theory of almost self-affine sets and measures. In: Fractals, Wavelets, and Their Applications. Springer Proceedings in Mathematics & Statistics, vol. 92, pp. 103–127. Springer (2014)

  35. 35.

    Tetenov, A.V.: Self-similar Jordan arcs and the graph directed systems of similarities. Sib. Math. J. 47(5), 940–949 (2006)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Viswanathan, P., Chand, A.K.B., Navascués, M.A.: Fractal perturbation preserving fundamental shapes: bounds on the scale factors. J. Math. Anal. Appl. 419(2), 804–817 (2014)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Viswanathan, P., Navascués, M.A., Chand, A.K.B.: Fractal polynomials and maps in approximation of continuous functions. Numer. Funct. Anal. Optim. 37(1), 106–127 (2016)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Wang, H.Y., Yu, J.S.: Fractal interpolation functions with variable parameters and their analytical properties. J. Approx. Theory 175, 1–18 (2013)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Zhang, K., Guo, S., Zhao, L., Zhao, X., Chan, H.L.W., Wang, Yu.: Realization of planar mixing by chaotic velocity in microfluidics. Microelectron. Eng. 88(6), 959–963 (2011)

    Article  Google Scholar 

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Acknowledgements

We would like to thank Dr. P. Viswanathan for his valuable comments during the preparations of this manuscript. The second author is thankful for the project: MTR/2017/000574 - MATRICS from the Science and Engineering Research Board (SERB), Government of India. We would like to thank anonymous reviewers for several constructive suggestions that greatly contributed to improve the paper.

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Correspondence to Sangita Jha.

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Jha, S., Chand, A.K.B. & Navascués, M.A. Approximation by shape preserving fractal functions with variable scalings. Calcolo 58, 8 (2021). https://doi.org/10.1007/s10092-021-00396-8

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Keywords

  • Fractals
  • Fractal interpolation functions
  • Smoothing
  • Shape preservation
  • Chebyshev system
  • Müntz approximation

Mathematics Subject Classification

  • 28A80
  • 41A15
  • 41A29
  • 41A30
  • 41A50
  • 42A15