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

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 = {ξs} s =1 n of n points in I, a sequence a = {as}s=1} n of n functions defined on n and a sequence Φn = {Vks} s=1 n 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=1 na s(f(ξ ),...,f(ξ))V{ks}. In searching an optimal sampling method, we study the quantity νn(f, Φ)q := {Φn, ξ, a}, ||F S(ξ, Φn, a, f)||q, where the infimum is taken over all sequences ξ = {ξ}s=1 n of n points, a = {as}s =1 n of n functions defined on n, and Φn = {Vks}}s=1 n of n functions from Φ. Let Up,θ be the unit ball in the Besov space B α p,θ} and M the set of centered B-spline wavelets Mk,s(x):= Nr(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 νn(U α p,θ, (f M)q:= sup f∞U α p,θ μn(f,M)q n α. An asymptotically optimal non-linear sampling recovery method S * for μn(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:n to C(I) and n fixed points ξ = {ξs} s=1 n. © 2008 Springer Science+Business Media, LLC. Author