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FRACTAL BASES FOR BANACH SPACES OF SMOOTH FUNCTIONS

Part of: Normed linear spaces and Banach spaces; Banach lattices Approximations and expansions Classical measure theory

Published online by Cambridge University Press:  30 July 2015

M. A. NAVASCUÉS ,
P. VISWANATHAN ,
A. K. B. CHAND ,
M. V. SEBASTIÁN  and
S. K. KATIYAR
M. A. NAVASCUÉS
Affiliation:
Departmento de Matemática Aplicada, Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza, C/- María de Luna 3, Zaragoza 50018, Spain email manavas@unizar.es
P. VISWANATHAN*
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, Australia email amritaviswa@gmail.com
A. K. B. CHAND
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai, India email chand@iitm.ac.in
M. V. SEBASTIÁN
Affiliation:
Centro Universitario de la Defensa de Zaragoza, Academia General Militar, Zaragoza, Spain email msebasti@unizar.es
S. K. KATIYAR
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai, India email sbhkatiyar@gmail.com
*
✉amritaviswa@gmail.com
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Abstract

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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.

Keywords

smooth fractal functionHermite fractal interpolation functionfractal operatortopological isomorphismSchauder basisMarkushevich basis

MSC classification

Primary: 28A80: Fractals
Secondary: 41A05: Interpolation 46B15: Summability and bases
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 3 , December 2015 , pp. 405 - 419
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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