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Local Fractal Interpolation on Unbounded Domains

Part of: Smooth dynamical systems: general theory Harmonic analysis in several variables Classical measure theory Approximations and expansions

Published online by Cambridge University Press:  23 January 2018

Peter R. Massopust
Peter R. Massopust*
Affiliation:
Centre of Mathematics, Research Unit M15, Technische Universität München, Boltzmannstrasse 3, 85747 Garching b. München, Germany ([email protected])
*
Corresponding author.
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Abstract

We define fractal interpolation on unbounded domains for a certain class of topological spaces and construct local fractal functions. In addition, we derive some properties of these local fractal functions, consider their tensor products, and give conditions for local fractal functions on unbounded domains to be elements of Bochner–Lebesgue spaces.

Keywords

local iterated function systemattractorfractal interpolationRead–Bajraktarević operatorfractal functionBochner–Lebesgue spaceunbounded componentnon-compact Hausdorff space

MSC classification

Primary: 28A80: Fractals 37C70: Attractors and repellers, topological structure 41A05: Interpolation 41A30: Approximation by other special function classes 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 151 - 167
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1. Adams, R. and Fourier, J., Sobolev spaces, 2nd edition (Academic Press, 2003).Google Scholar
2. Barnsley, M. F., Fractal functions and interpolation, Constr. Approx. 2 (1986), 303–329.Google Scholar
3. Barnsley, M. F., Fractals everywhere (Dover Publications, New York, 2012).Google Scholar
4. Barnsley, M. F. and Demko, S., Iterated function systems and the global construction of fractals, Proc. R. Soc. Lond. A 399 (1985), 243–275.Google Scholar
5. Barnsley, M. F. and Hurd, L. P., Fractal image compression (AK Peters Ltd, Wellesley, MA, 1993).Google Scholar
6. Barnsley, M. F. and Massopust, P., Bilinear fractal interpolation and box dimension, J. Approx. Theory 192 (2015), 363–378.CrossRefGoogle Scholar
7. Barnsely, M. F., Hegland, M. and Massopust, P., Numerics and fractals, Bull. Inst. Math. Acad. Sin. (N.S.) 9(3) (2014), 389–430.Google Scholar
8. Berlanga, R. and Epstein, D. B. A., Measures on sigma-compact manifolds and their equivalence under homeomorphisms, J. Lond. Math. Soc. 27(2) (1983), 63–74.Google Scholar
9. Geoghegan, R., Topological methods in group theory (Springer, New York, 2008).Google Scholar
10. Geronimo, J., Hardin, D. and Massopust, P., Fractal functions and wavelets expansions based on several scaling functions, J. Approx. Theory 78(3) (1994), 373–401.CrossRefGoogle Scholar
11. Hutchinson, J. E., Fractals and self similarity, Indiana Univ. J. Math. 30 (1981), 713–747.Google Scholar
12. Klee, V., Connectedness in topological linear spaces, Israel J. Math. 2(2) (1964), 127–131.Google Scholar
13. Massopust, P. R., Fractal functions and their applications, Chaos Solitons Fractals 8(2) (1997), 171–190.Google Scholar
14. Massopust, P. R., Interpolation with splines and fractals (Oxford University Press, New York, 2012).Google Scholar
15. Massopust, P. R., Local fractal functions and function spaces, Springer Proc. Math. Stat. 92 (2014), 245–270.Google Scholar
16. Massopust, P. R., Local fractal functions in Besov and Triebel–Lizorkin spaces, J. Math. Anal. Appl. 436 (2016), 393–407.Google Scholar
17. Massopust, P. R., Fractal functions, fractal surfaces, and wavelets, 2nd edition (Academic Press, 2016).Google Scholar
18. Rudin, W., Functional analysis (McGraw-Hill, New York, 1991).Google Scholar