We investigate a problem of approximate non-linear sampling recovery of functions on the interval 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 I, we choose a sequence ξ = {ξ^s}_{s =1}ⁿ of n points in I, a sequence a = {a_s}_s=1}ⁿ of n functions defined on ℝⁿ and a sequence Φ_n = {V_ks} _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, Φ_n,a,f):= ∑_s=1ⁿa _s(f(ξ¹),...,f(ξⁿ))V{k_s}. In searching an optimal sampling method, we study the quantity ν_n(f, Φ)_q := {Φ_n, ξ, a}, ||F - S(ξ, Φ_n, a, f)||_q, where the infimum is taken over all sequences ξ = {ξ^s}_s=1ⁿ of n points, a = {a_s}_{s =1}ⁿ of n functions defined on ℝⁿ, and Φn = {V_ks}}_s=1ⁿ of n functions from Φ. Let U^α_p,θ be the unit ball in the Besov space B^α_p,θ} and M the set of centered B-spline wavelets M_k,s(x):= N_r(2^k x + ρ - s), which do not vanish identically on I , where N _r is the B-spline of even order r ≥ [α] + 1 with knots at the points 0,1,...,r. For 1 ≤ p,q ≤ ∞, 0 < θ ≤ ∞ and α > 1, we proved the following asymptotic order ν_n(U ^α_p,θ, (f M)_q:= sup _{f∞U α p,θ} μ_n(f,M)_q n ^{- α}. An asymptotically optimal non-linear sampling recovery method S ^* for μ_n(U^α_p,θ, (f 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 to C(I) and n fixed points ξ = {ξ^s} _s=1ⁿ. © 2008 Springer Science+Business Media, LLC.

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