Stability of wavelet frames with matrix dilations

Ole Christensen, Wenchang Sun

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    Under certain assumptions we show that a wavelet frame

    {Tau(A(j), b(j,k))psi} (j,k is an element of Z) := {vertical bar detA(j)vertical bar(-1/2) psi(A(j)(-1)(x - b(j,k)))} (j,k is an element of Z)

    in L-2(R-d) remains a frame when the dilation matrices A(j) and the translation parameters b(j,k) are perturbed. As a special case of our result, we obtain that if {Tau(A(j), A(j)Bn)psi} (j is an element of Z, n is an element of Zd) is a frame for an expansive matrix A and an invertible matrix B, then {Tau(A'(j), A(j)B lambda(n))psi}(j is an element of Z,) (n is an element of) (Zd) is a frame if vertical bar vertical bar A(-j)A'(j) - I vertical bar vertical bar(2) <= epsilon and vertical bar vertical bar lambda(n) - n vertical bar vertical bar infinity <= eta for sufficiently small epsilon,eta > 0.
    Original languageEnglish
    JournalProceedings of the American Mathematical Society
    Volume134
    Issue number3
    Pages (from-to)831-842
    ISSN0002-9939
    DOIs
    Publication statusPublished - 2006

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