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

• Main Author: Dinh Dũng
• Format: Book Article Dataset
• Language: Vietnamese
• Published: Journal of Complexity 2016
• Subjects: Quasi-interplant; Quasi-interpolant representation; Mixed B-spline; Besov space of mixed smoothness
• Online Access: http://repository.vnu.edu.vn/handle/VNU_123/10978
• Institution: Vietnam National University, Hanoi

Let be a set of n sample points in the d-cube Id≔[0,1]d, and a family of n functions on Id. We define the linear sampling algorithm Ln(Φ,ξ,⋅) for an approximate recovery of a continuous function f on Id from the sampled values f(x1),…,f(xn), by For the Besov class of mixed smoothness α, to study optimality of Ln(Φ,ξ,⋅) inLq(Id) we use the quantity where the infimum is taken over all sets of n sample points and all families in Lq(Id). We explicitly constructed linear sampling algorithms Ln(Φ,ξ,⋅)on the set of sample points ξ=Gd(m)≔{(2−k1s1,…,2−kdsd)∈Id:k1+⋯+kd≤m}, with Φ 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 r. For various 0<p,q,θ≤∞ and 1/p<α<r, we proved upper bounds for the worst case error which coincide with the asymptotic order of in some cases. A key role in constructing these linear sampling algorithms, plays a quasi-interpolant representation of functions by mixed B-spline series with the coefficient functionals which are explicitly constructed as linear combinations of an absolute constant number of values of functions. Moreover, we proved that the quasi-norm of the Besov space is equivalent to a discrete quasi-norm in terms of the coefficient functionals.