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The Construction of G-Invariant Finite Tight Frames

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Abstract

We give a complete classification of the finite tight frames which are G-invariant, i.e., invariant under the unitary action of a group G. This result is constructive, and we use it to consider a number of examples. In particular, we determine the minimum number of generators for a tight frame for the orthogonal polynomials on an n-gon or cube, which is invariant under the symmetries of the weight.

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Acknowledgments

We would like to thank Steven Sam for pointing out that for a general complex reflection group G, (4.15) describes how \(S_N\) decomposes as a sum of irreducible representations of G.

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Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand

    Richard Vale & Shayne Waldron

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  1. Richard Vale
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  2. Shayne Waldron
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Correspondence to Shayne Waldron.

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Communicated by Peter G. Casazza.

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Vale, R., Waldron, S. The Construction of G-Invariant Finite Tight Frames. J Fourier Anal Appl 22, 1097–1120 (2016). https://doi.org/10.1007/s00041-015-9443-9

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  • DOI: https://doi.org/10.1007/s00041-015-9443-9

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