• Admin
    UC Research Repository
    View Item 
       
    • UC Home
    • Library
    • UC Research Repository
    • College of Engineering
    • Engineering: Reports
    • View Item
       
    • UC Home
    • Library
    • UC Research Repository
    • College of Engineering
    • Engineering: Reports
    • View Item
    JavaScript is disabled for your browser. Some features of this site may not work without it.

    Browse

    All of the RepositoryCommunities & CollectionsBy Issue DateAuthorsTitlesSubjectsThis CollectionBy Issue DateAuthorsTitlesSubjects

    Statistics

    View Usage Statistics

    Spanning and sampling in Lebesgue and Sobolev spaces

    Thumbnail
    View/Open
    bui_laugesen_ucdms2004-8_report.pdf (6.689Mb)
    Author
    Bui, Huy-Qui
    Laugesen, R. S.
    Date
    2004
    Permanent Link
    http://hdl.handle.net/10092/12719

    We establish conditions under which the small-scale affine system {𝜓(𝑎j 𝑥 - 𝑘) : j ≥ J, 𝑘 ∈โ„คแตˆ}, with J∈โ„ค fixed, spans the Lebesgue space 𝐿แต–(โ„แตˆ) and the Sobolev space 𝑊แต,แต–(โ„แตˆ), 1 ≤ 𝑝 < ∞. The dilation matrices 𝑎j are expanding (meaning limj →∞ ll𝑎j โป¹ ll = 0) but they need not be diagonal. For spanning 𝐿แต– we require ∫โ„แตˆ 𝜓𝑑𝑥 ≠ 0 and (when 𝑝 > 1) that the periodization of l𝜓l or of 𝟙{𝜓≠โ‚€} be bounded. To span 𝑊แต,แต– we also require a Strang-Fix condition on 𝜓. But we impose this condition only to order 𝑚 - 1, whereas earlier authors required order 𝑚. Our spanning results are derived from explicit "Shannon" type sampling formulas that express an arbitrary function 𝑓 as a limit of linear combinations of the 𝜓(𝑎j 𝑥 - 𝑘). The coefficients in these sampling formulas are local averages of 𝑓, or pointwise values off when 𝑓 has some regularity.

    Subjects
    Sampling
     
    spanning
     
    completeness
     
    density
     
    Strang-Fix
     
    approximation order
     
    quasi-interpolation
     
    approximate approximations
     
    wavelet
     
    Field of Research::01 - Mathematical Sciences
    Collections
    • Engineering: Reports [684]
    Rights
    http://library.canterbury.ac.nz/thesis/etheses_copyright.shtml

    UC Research Repository
    University Library
    University of Canterbury
    Private Bag 4800
    Christchurch 8140

    Phone
    364 2987 ext 8718

    Email
    ucresearchrepository@canterbury.ac.nz

    Follow us
    FacebookTwitterYoutube

    © University of Canterbury Library
    Send Feedback | Contact Us