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On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle

  Published:2013-07-19
 Printed: Jun 2014
  • Ole Christensen,
    Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Lyngby, Denmark
  • Hong Oh Kim,
    Department of Mathematical Sciences, KAIST Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea
  • Rae Young Kim,
    Department of Mathematics, Yeungnam University, Dae-dong, Gyeongsan-si, Gyeongsangbuk-do, 712-749, Republic of Korea
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Abstract

The unitary extension principle (UEP) by Ron and Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators, and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.
Keywords: Bessel sequences, frames, extension of wavelet Bessel system to tight frame, wavelet systems, unitary extension principle Bessel sequences, frames, extension of wavelet Bessel system to tight frame, wavelet systems, unitary extension principle
MSC Classifications: 42C15, 42C40 show english descriptions General harmonic expansions, frames
Wavelets and other special systems
42C15 - General harmonic expansions, frames
42C40 - Wavelets and other special systems
 

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