Nonnegative Scaling Vectors on the Interval

Author： Ruch David K. Van Fleet Patrick J.

Publisher： MDPI

E-ISSN： 2075-1680|2|3|371-389

ISSN： 2075-1680

Source： Axioms, Vol.2, Iss.3, 2013-07, pp. : 371-389

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Abstract

In this paper, we outline a method for constructing nonnegative scaling vectors on the interval. Scaling vectors for the interval have been constructed in [1,2,3]. The approach here is different in that the we start with an existing scaling vector Φ that generates a multi-resolution analysis for L 2 ( R ) to create a scaling vector for the interval. If desired, the scaling vector can be constructed so that its components are nonnegative. Our construction uses ideas from [4,5] and we give results for scaling vectors satisfying certain support and continuity properties. These results also show that less edge functions are required to build multi-resolution analyses for L 2 [ a , b ] than the methods described in [5,6].