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

• Authors: Dinh Dng
• Venue: Journal of Complexity
• Year: 2011

Let @x={x^j}"j"="1^n be a set of n sample points in the d-cube I^d@?[0,1]^d, and @F={@f"j}"j"="1^n a family of n functions on I^d. We define the linear sampling algorithm L"n(@F,@x,@?) for an approximate recovery of a continuous function f on I^d from the sampled values f(x^1),...,f(x^n), by L"n(@F,@x,f)@?@?j=1nf(x^j)@f"j. For the Besov class B"p","@q^@a of mixed smoothness @a, to study optimality of L"n(@F,@x,@?) in L"q(I^d) we use the quantity r"n(B"p","@q^@a)"q@?inf@x,@Fsupf@?Bp,@q@a@?f-L"n(@F,@x,f)@?"q, where the infimum is taken over all sets of n sample points @x={x^j}"j"="1^n and all families @F={@f"j}"j"="1^n in L"q(I^d). We explicitly constructed linear sampling algorithms L"n(@F,@x,@?) on the set of sample points @x=G^d(m)@?{(2^-^k^"^1s"1,...,2^-^k^"^ds"d)@?I^d:k"1+...+k"d@?m}, with @F 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