Bivariate Interpolation by Splines and Approximation Order

We construct Hermite interpolation sets for bivariate spline spaces of arbitrary degree and smoothness one on non-rectangular domains with uniform type triangulations. This is done by applying a general method for constructing Lagrange interpolation sets for bivariate spline spaecs of arbitrary degree and smoothness. It is shown that Hermite interpolation yields (nearly) optimal approximation order. Applications to data fitting problems and numerical examples are given

Strong unicity in nonlinear approximation and free knot splines

We give a necessary alternation condition for unique local best approximation from Sm,k, the set of splines of degree m with k free knots. This result is related to a conjecture of L.L. Schumaker. Moreover, we give a characterization of functions from the interior of the strong unicity set for S1m,k, the set of splines of degree m with k free simple knots, and show that this set is dense in the unicity set. Then we give a general characterization of suns for strong unicity and show that S1m,k is a set of this type, although it is not a sun

Strong unicity of best approximations : a numerical aspect

The set of functions in C(T) which have a strongly unique best approximation from a given finite-dimensional subspace is denoted by SU(G). Since strong unicity plays an important role in numerical computations and since there the functions are only known up to some error, it is natural to ask what are the functions from the interior of SU(G). A complete characterization of those functions is given and the result is applied to weak Chebyshev and spline subspaces

Bivariate Segment Approximation

In this note we state some problems on approximation by univariate splines with free knots, bivariate segment approximation and tensor product splines with variable knot lines. There is a vast literature on approximation and interpolation by univariate splines with fixed knots (see e.g. the books of de Boor [1], Braess [2], DeVore & Lorentz [4], Powell [20], Schumaker [21], Nürnberger [13] and the book of Chui [3] on multivariate splines). On the other hand, numerical examples show that in general, the error is much smaller if variable knots are used for the approximation of functions instead of fixed knots. This is true for univariate splines as well as for bivariate splines. But approximation by splines with free knots leads to rather difficult nonlinear problems.[...

The metric projection for free knot splines

Only few results are known on continuity properties of the set-valued metric projection in nonlinear uniform approximation. In this paper we investigate this mapping in the case of best uniform approximation by splines of degree m with k free knots. A characterization of those functions at which the metric projection is upper semicontinuous is given. It follows that the metric projection is upper semicontinuous if and only if k ≤ m, and that it is upper semicontinuous at all "normal" functions. On the other hand, it is shown that the metric projection is never lower semicontinuous

Interpolation by spline spaces on classes of triangulations

We describe a general method for constructing triangulations Δ which are suitable for interpolation by Srq(Δ),

Interpolation by C1 Splines of Degree q 4 on Triangulations

Let Δ be an arbitrary regular triangulation of a simply connected compact polygonal domain Ω ⊂ IR² and let S1q (Δ) denote the space of bivariate polynomial splines of degree q and smoothness 1 with respect to Δ. We develop an algorithm for constructing point sets admissible for Lagrange interpolation by S1q (Δ) if q ≥ 4. In the case

Bivariate Segment Approximation and Splines

The problem to determine partitions of a given rectangle which are optimal for segment approximation (e.g. by bivariate piecewise polynomials) is investigated. We give criteria for optimal partitions and develop algorithms for computing optimal partitions of certain types. It is shown that there is a surprising relationship between various types of optimal partitions. In this way, we obtain good partitions for interpolation by tensor product spline spaces. Our numerical examples show that the methods work efficiently

Bivariate Spline Interpolation with Optimal Approximation Order

Let Δ be a triangulation of some polygonal domain Δ ⊂ R² and let Srq(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We present a Hermite type interpolation scheme for Srq(Δ), q ≥ 3r +2, that possesses optimal approximation order Ο(h q+1). Furthermore, the fundamental functions of our scheme form a locally linearly independent basis for a superspline subspace of Srq(Δ)