We consider the approximation (either interpolation, or least-squares) of L functions in the shift-invariant space VT = spank∈Z{φ( t T − n)} that is generated by the single shifted function φ. We measure the approximation error in an L sense and evaluate the asymptotic equivalent of this error as the sampling step T tends to zero. Let f ∈ L and fT be its approximation in VT . It is well-known t...

Best parametric spline interpolation extends and refines the classical spline problem of best interpolation to (∗) inf t inf f { ∫ 1 0 ‖f (t)‖dt : f(ti) = yi, 1 ≤ i ≤ n} Here t : 0 = t1 < . . . < tn = 1 denotes a sequence of nodes and yi data in R d with y i 6= y i+1 . The R valued functionsf(t) lie componentwise in the Sobolev space L2(0, 1) and ‖ ‖ denotes the Euclidean norm in R. This proble...

The paper is related to the article It is proved that a sequence of L spline approximations of a given continuous function f in the interval a b converges in this interval uniformly to f cf A sequence of L spline approximations of a given function f in the interval a b In the interval a b of nite length L b a let us consider a real function x f x Let n be a posi tive integer We divide the inter...

A scheme to specify planar C2 Pythagorean-hodograph (PH) quintic spline curves by control polygons is proposed, in which the “ordinary” C2 cubic B-spline curve serves as a reference for the shape of the PH spline. The method facilitates intuitive and efficient constructions of open and closed PH spline curves, that typically agree closely with the corresponding cubic B-spline curves. The C2 PH ...

Spline methods provide an important tool to solve singularly perturbed boundary value problems. The present paper gives a comprehensive review of computational methods based on splines used in the solution of various classes of singularly perturbed problems such as self adjoint, linear, non-linear, semi-linear, quasi-linear, singular, single parameter and multi parameter. The spline based numer...