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On the Completeness and Other Properties of Some Function Systems inLp, 0<p<∞

✍ Scribed by Vadim I Filippov

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
292 KB
Volume
94
Category
Article
ISSN
0021-9045

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✦ Synopsis

We study minimal conditions under which a function system is a representation system in L p (a, b), & a1, b>0, and . is an arbitrary function from L p , arose in connection with investigations of wavelets [BDR, D, JL, JM, Ml, Md, Mey] and frames [ChSh1, ChSh2] and in connection with questions of image compression. The questions of completeness of integer translates in function spaces on R are considered in [AO]. Moreover, the authors cited above and others continue to attempt to generalize the classical systems (Haar, trigonometric, Faber Schauder, etc.) to get systems with new properties.
Since there exist an infinite number of function systems, the question arises: ``How can we find the optimal function system for a given problem?'' To answer such a question we should somehow classify all such function systems.
Properties of function systems depend on the function spaces (for example; there is no basis in L p (0, 1), 0< p<1). Therefore, it is natural to consider function systems in the spaces: (1) L p (0, 1), 0< p<1, (2) L p (0, 1), 1 p< (if the field of investigation is limited by the scale of L p , p>0).
The notion of a representation system which generalizes the notion of a basis was introduced by A. A. Talalyan [T1]. There are investigations on subsystems of representation systems [I, F1, F2], on representation systems in Article No. AT983176 42

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