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Generation of finite tight frames by Householder transformations

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Abstract

Finite tight frames are widely used for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such a purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting of m vectors in ℝn or ℂn only O(nm) operations are needed. In addition, we also study the following question: Given a set of vectors in ℝn or ℂn, how many additional vectors, possibly with constraints, does one need to add in order to obtain a tight frame?

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China

    De-Jun Feng

  2. Mathematics Department, Southern Polytechnic State University, Mariieta, GA, 30060, U.S.A.

    Long Wang

  3. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, U.S.A.

    Yang Wang

Authors
  1. De-Jun Feng
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  2. Long Wang
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  3. Yang Wang
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Corresponding author

Correspondence to De-Jun Feng.

Additional information

Communicated by J. Benedetto

Dedicated to Dr. Micchelli for his 60th birthday

Mathematics subject classification (2000)

Primary 42C15.

De-Jun Feng: Supported in part by the Special Funds for Major State Basic Research Projects in China.

Yang Wang: Supported in part by the National Science Foundation, grants DMS-0070586 and DMS-0139261.

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Feng, DJ., Wang, L. & Wang, Y. Generation of finite tight frames by Householder transformations. Adv Comput Math 24, 297–309 (2006). https://doi.org/10.1007/s10444-004-7637-9

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  • DOI: https://doi.org/10.1007/s10444-004-7637-9

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