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In this paper we consider the problem of designing piecewise polynomial local interpolants of non-uniformly spaced data. We provide a constructive approach that, for any assigned degree of polynomial reproduction, continuity order, and support width, allows for generating the fundamental spline functions of minimum degree having the desired properties. Finally, the proposed construction is extended to handle open sets of data and to the case of multiple knots. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a general framework for the construction of piecewise-polynomial local interpolants with given smoothness and approximation order, defined on non-uniform knot partitions. We design such splines through a suitable combination of polynomial interpolants with either polynomial or rational, compactly supported blending functions. In particular, when the blending functions are rational, our approach provides spline interpolants having low, and sometimes minimum degree. Thanks to its generality, the proposed framework also allows us to recover uniform local interpolating splines previously proposed in the literature, to generalize them to the non-uniform case, and to complete families of arbitrary support width. Furthermore it provides new local interpolating polynomial splines with prescribed smoothness and polynomial reproduction properties. [Copyright &y& Elsevier]

In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots-that are not considered by the authors of the corresponding spline basis-thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data. [ABSTRACT FROM AUTHOR]

Abstract: Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that the most convenient parameter values may be chosen as well as the intervals for insertion. Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control. [Copyright &y& Elsevier]

Degree elevation is a typical corner-cutting algorithm. It refers to the process transforming control polygons when embedding a polynomial space of some degree into any polynomial space of higher degree. Dimension elevation similarly refers to the transformation of control polygons when embedding an Extended Chebyshev space possessing a Bernstein basis into another one, of higher dimension. Unlike degree elevation, this cannot always be split into successive (corner-cutting) steps elevating the dimension by one. What happens when it is not possible is investigated here. We shall see that the new control points can even be located outside the initial control polygons, giving evidence that dimension elevation is not always corner-cutting. [ABSTRACT FROM AUTHOR]