Sobolev spaces and approximation by affine spanning systems.

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dc.contributor.authorBui, H.-Q.
dc.contributor.authorLaugesen, R. S.
dc.date.accessioned2015-11-30T20:46:05Z
dc.date.available2015-11-30T20:46:05Z
dc.date.issued2006en
dc.description.abstractWe develop conditions on a Sobolev function 𝜓∈Wm,p(ℝd) such that if 𝜓̂ (0) = 1 and 𝜓 satisfies the Strang-Fix conditions to order m - 1, then a scale averaged approximation formula holds for all 𝑓 ∈ Wm,p(ℝd): [FORMULA] The dilations {𝑎j } are lacunary, for example 𝑎j = 2j, and the coefficients Cj,k are explicit local averages of 𝑓, or even pointwise sampled values, when 𝑓 has some smoothness. For convergence just in Wm-1,p(ℝd) the scale averaging is unnecessary and one has the simpler formula 𝑓(x) = limj→∞ ∑k∈ℤd cj,k 𝜓(𝑎jx-k). The Strang-Fix rates of approximation are recovered. As a corollary of the scale averaged formula, we deduce new density or "spanning" criteria for the small scale affine system { 𝜓(𝑎jx-k) : j > 0, k ∈ ℤd } in Wm,p(ℝd). We also span Sobolev space by derivatives and differences of affine systems, and we raise an open problem: does the Gaussian affine system span Sobolev space?
dc.identifier.urihttp://hdl.handle.net/10092/11451
dc.language.isoen
dc.publisherUniversity of Canterbury. Dept. of Mathematics and Statisticsen
dc.relation.isreferencedbyNZCU
dc.rightsCopyright H.-Q. Buien
dc.rights.urihttps://canterbury.libguides.com/rights/theses
dc.subjectCompletenessen
dc.subjectquasi-interpolationen
dc.subjectStrang-Fixen
dc.subjectapproximate identityen
dc.subjectscale averagingen
dc.subject.anzsrcField of Research::01 - Mathematical Sciences::0102 - Applied Mathematicsen
dc.titleSobolev spaces and approximation by affine spanning systems.en
dc.typeDiscussion / Working Papers
uc.collegeFaculty of Engineering
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