FRACTAL BASES FOR BANACH SPACES OF SMOOTH FUNCTIONS

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to...

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Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society Vol. 92; no. 3; pp. 405 - 419
Main Authors: NAVASCUÉS, M. A., VISWANATHAN, P., CHAND, A. K. B., SEBASTIÁN, M. V., KATIYAR, S. K.
Format: Journal Article
Language:English
Published: Cambridge, UK Cambridge University Press 01.12.2015
Subjects:
Banach spaces
secondary 41A05
fractal operator
Hermite fractal interpolation function
46B15
Markushevich basis
topological isomorphism
primary 28A80
smooth fractal function
Schauder basis
ISSN:0004-9727, 1755-1633
Online Access:Get full text
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Summary:This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called $\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space ${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for ${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.
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ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972715000738