Sharp Estimates of the Widths of Convolution Classes

Generalised sk-Spline Interpolation on Compact Abelian Groups

“…The sk-spline is a recent generalisation of the more familiar polynomial splines (which are realised when k is a Bernoulli monospline of an appropriate degree) and have proved effective for computing the n-widths of certain convolution classes of functions [9,10]. The construction in (1) has been used by many authors in the case when the kernel k possesses some sign regularity properties (like cyclic variation diminishing), as well as using Taylor's Theorem to provide some error estimates [4,19,20,21,22,24,27].…”

Optimal Subspaces forn-Widths of Bounded Subsets in Hilbert Spaces

“…In [14,15], the n-widths d n (A; H ) with odd number n were obtained for the cases when A = W r , the space of functions whose (r − 1)th derivative was absolutely continuous and r th derivative satisfied the inequality 2 0 |f (r )(x)|dx ≤ 1, and H = L(0, 2 ), the space of summable functions of period 2 . For the n-widths and optimal subspaces involving some convolution classes and L-splines, where L was an linear differential operator of nth order with constant real coefficients, the results were obtained by [13] and by [9].…”

“…In [11], 652 Q. H. Li we developed the generalized L-spline spaces to approximate the solution of problems with rough coefficient functions, where L was a linear differential operator with bounded measurable coefficient functions. The special basis functions of the generalized L-spline space could be viewed as a generalization of the L-spline functions as in [9,13], where L was an linear differential operator with constant real coefficients. Furthermore, they reflected the property of the unknown solution of problems with rough data.…”

“…Let L and L * be defined in (9) and (10). Let H = H m (a, b) and V = u ∈ H m (a, b), L * Lu| I j ∈ L 2 (I j ), for j = 1, 2, , n .…”

Interpolation on the Torus using sk-Splines with Number Theoretic Knots

“…Such functions are natural generalization of periodic polynomial splines, realized when K is a Bernoulli monospline of appropriate order. The sk-splines were introduced, and their basic theory developed, by Kushpel [6,7]. In this paper we continue with the development of error estimates for sk-spline interpolation, begun, in the univariate case, in [8], and extended into higher dimensions in [4,11].…”