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Non-linear sampling recovery based on quasi-interpolant wavelet representations

Dinh Dũng

http://repository.vnu.edu.vn/handle/VNU_123/10993

Truy cập trực tuyến

  • Nhan đề:
    Non-linear sampling recovery based on quasi-interpolant wavelet representations
  • Tác giả: Dinh Dũng
  • Chủ đề: Adaptive choice; B-spline; Besov space; Non-linear sampling recovery; Quasiinterpolant wavelet representation
  • Mô tả: We investigate a problem of approximate non-linear sampling recovery of functions on the interval I:=[0,1]I:=[0,1] expressing the adaptive choice of n sampled values of a function to be recovered, and of n terms from a given family of functions Φ. More precisely, for each function f on II, we choose a sequence ξ={ξs}ns=1ξ={ξs}s=1n of n points in II, a sequence a={as}ns=1a={as}s=1n of n functions defined on RnRnand a sequence Φn={φks}ns=1Φn={φks}s=1n of n functions from a given family Φ. By this choice we define a (non-linear) sampling recovery method so that f is approximately recovered from the n sampled values f(ξ 1), f(ξ 2),..., f(ξ n ), by the n-term linear combination S(f)= S(ξ,Φn,a,f):= ∑s=1nas(f(ξ1),...,f(ξn))φks.S(f)= S(ξ,Φn,a,f):= ∑s=1nas(f(ξ1),...,f(ξn))φks. In searching an optimal sampling method, we study the quantity νn(f,Φ)q:= infΦn,ξ,a∥f−S(ξ,Φn,a,f)∥q,νn(f,Φ)q:= infΦn,ξ,a∥f−S(ξ,Φn,a,f)∥q, where the infimum is taken over all sequences ξ={ξs}ns=1ξ={ξs}s=1n of n points, a={as}ns=1a={as}s=1n of nfunctions defined on RnRn, and Φn={φks}ns=1Φn={φks}s=1n of n functions from Φ. Let Uαp,θUp,θα be the unit ball in the Besov space Bαp,θ,Bp,θα, and M the set of centered B-spline wavelets Mk,s(x):= Nr(2kx+ρ−s),Mk,s(x):= Nr(2kx+ρ−s), which do not vanish identically on II, where N r is the B-spline of even order r = 2ρ ≥ [α] + 1 with knots at the points 0,1,...,r. For 1≤p,q≤∞, 0<θ≤∞1≤p,q≤∞, 0<θ≤∞ and α > 1, we proved the following asymptotic order νn(Uαp,θ,M)q := supf∈Uαp,θνn(f,M)q ≍ n−α.νn(Up,θα,M)q := supf∈Up,θανn(f,M)q ≍ n−α. An asymptotically optimal non-linear sampling recovery method S * for νn(Uαp,θ,M)qνn(Up,θα,M)q is constructed by using a quasi-interpolant wavelet representation of functions in the Besov space in terms of the B-splines M k,s and the associated equivalent discrete quasi-norm of the Besov space. For 1 ≤ p < q ≤ ∞ , the asymptotic order of this asymptotically optimal sampling non-linear recovery method is better than the asymptotic order of any linear sampling recovery method or, more generally, of any non-linear sampling recovery method of the form R(H,ξ,f): = H(f(ξ 1),...,f(ξ n )) with a fixed mapping H:Rn→C(I)H:Rn→C(I) and n fixed points ξ={ξs}ns=1.
  • Nơi xuất bản: Advances in Computational Mathematics
  • Năm xuất bản: 2009
  • Số nhận dạng: http://repository.vnu.edu.vn/handle/VNU_123/10993

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