The inequality of Erdos-Turan-Koksma. An arbitrary complete orthonormal functional system on B-ADIC base

Grozdanov, V.; Stoilova, S.

Dokladi na B"lgarskata Akademiya na Naukite 57(5): 23-28

2004


Accession: 012660455

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Abstract
In the present paper the authors consider an arbitrary complete orthonormal functional system constructed in a generalized number system with some additional conditions for the values of the functions of this system as a tool for studying discrepancy. Estimations of the extreme discrepancy and the star-discrepancy of an arbitrary s-dimensional net are shown in explicit form in the terms of the sum of this net with respect to the functions of this system. The obtained estimations of the extreme discrepancy and the star-discrepancy are analogies of the classical inequality of Erdos-Turan-Koksma. As consequences of the main result we obtain estimations of the extreme discrepancy of an arbitrary s-dimensional net in the terms of sums of this net with respect to the functions of some well-known orthonormal functional systems - the Price functional system, the Walsh functional system, the Walsh functional system over finite groups and the Vilenkin-like functional system.