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INFORMATICA, 2014, Vol. 25, No. 4, 643-662
© Institute of Mathematics and Informatics,
DOI: http://dx.doi.org/10.15388/Informatica.2014.33

ISSN 0868-4952

Atomic Decompositions of Fuzzy Normed Linear Spaces for Wavelet Applications

Sorin NADABAN, Ioan DZITAC1

Department of Mathematics and Computer Science Aurel Vlaicu University of Arad, Elena Dragoi 2, RO-310330 Arad, Romania Department of Social Sciences, Agora University of Oradea Piata Tineretului 8, RO-485526 Oradea, Romania E-mail: sorin.nadaban@uav.ro, ioan.dzitac@uav.ro, idzitac@univagora.ro, rector@univagora.ro

Abstract

Wavelet analysis is a powerful tool with modern applications as diverse as: image processing, signal processing, data compression, data mining, speech recognition, computer graphics, etc. The aim of this paper is to introduce the concept of atomic decomposition of fuzzy normed linear spaces, which play a key role in the development of fuzzy wavelet theory. Atomic decompositions appeared in applications to signal processing and sampling theory among other areas.

First we give a general definition of fuzzy normed linear spaces and we obtain decomposition theorems for fuzzy norms into a family of semi-norms, within more general settings. The results are both for Bag-Samanta fuzzy norms and for Katsaras fuzzy norms. As a consequence, we obtain locally convex topologies induced by this types of fuzzy norms.

The results established in this paper, constitute a foundation for the development of fuzzy operator theory and fuzzy wavelet theory within this more general frame.

Keywords:

fuzzy wavelet, atomic decomposition, fuzzy metric space, fuzzy norm, fuzzy normed linear space (FNLS)


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