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A Fractal Operator on Some Standard Spaces of Functions
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 60 / Issue 3 / August 2017
- Published online by Cambridge University Press:
- 10 January 2017, pp. 771-786
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Through appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as an α-fractal operator on
, the space of all real-valued continuous functions defined on a compact interval I. With an eye towards connecting fractal functions with other branches of mathematics, in this paper we continue to investigate the fractal operator in more general spaces such as the space 
of all bounded functions and the Lebesgue space 
, and in some standard spaces of smooth functions such as the space 
of k-times continuously differentiable functions, Hölder spaces 
and Sobolev spaces 
. Using properties of the α-fractal operator, the existence of Schauder bases consisting of self-referential functions for these function spaces is established.
, the space of all real-valued continuous functions defined on a compact interval 
of all bounded functions and the Lebesgue space 
, and in some standard spaces of smooth functions such as the space 
of 
and Sobolev spaces 
. Using properties of the 