Extension of shift-invariant systems in L2(ℝ) to frames

Marcin Bownik, Ole Christensen, Xinli Huang, Baiyun Yu

    Research output: Contribution to journalJournal articleResearchpeer-review


    In this article, we show that any shift-invariant Bessel sequence with an at most countable number of generators can be extended to a tight frame for its closed linear span by adding another shift-invariant system with at most the same number of generators. We show that in general this result is optimal, by providing examples where it is impossible to obtain a tight frame by adding a smaller number of generators. An alternative construction (which avoids the technical complication of extracting the square root of a positive operator) yields an extension of the given Bessel sequence to a pair of dual frame sequences. © 2012 Taylor and Francis Group, LLC.
    Original languageEnglish
    JournalNumerical Functional Analysis and Optimization
    Issue number7-9
    Pages (from-to)833-846
    Publication statusPublished - 2012


    • Functional analysis
    • Mathematical techniques
    • Invariance


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