B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness

Let ξ={xj}j=1n be a set of nn sample points in the dd-cube Id≔[0,1]dId≔[0,1]d, and Φ={φj}j=1n a family of nn functions on IdId. We define the linear sampling algorithm Ln(Φ,ξ,⋅)Ln(Φ,ξ,⋅) for an approximate recovery of a continuous function ff on IdId from the sampled values f(x1),…,f(xn)f(x1),…,f(xn), by Ln(Φ,ξ,f)≔∑j=1nf(xj)φj. For the Besov class Bp,θα of mixed smoothness αα, to study optimality of Ln(Φ,ξ,⋅)Ln(Φ,ξ,⋅) in Lq(Id)Lq(Id) we use the quantity rn(Bp,θα)q≔infξ,Φsupf∈Bp,θα‖f−Ln(Φ,ξ,f)‖q, where the infimum is taken over all sets of nn sample points ξ={xj}j=1n and all families Φ={φj}j=1n in Lq(Id)Lq(Id). We explicitly constructed linear sampling algorithms Ln(Φ,ξ,⋅)Ln(Φ,ξ,⋅) on the set of sample points ξ=Gd(m)≔{(2−k1s1,…,2−kdsd)∈Id:k1+⋯+kd≤m}ξ=Gd(m)≔{(2−k1s1,…,2−kdsd)∈Id:k1+⋯+kd≤m}, with ΦΦ a family of linear combinations of mixed B-splines which are mixed tensor products of either integer or half integer translated dilations of the centered B-spline of order rr. For various 0