In this paper we consider discrete splines S(j), j ∈ Z, with equidistant nodes which may grow as O(|j|s) as |j| → ∞. Such splines are relevant for the purposes of digital signal processing. We give the definition of discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete sp...

Quadratic Powell-Sabin splines and their rational extension, the socalled NURPS surfaces, are an interesting alternative for classical tensor-product NURBS in the context of isogeometric analysis, because they allow the use of local refinements while retaining a Bspline like representation and exact description of conic sections. In this paper we present a simple and effective strategy to conve...

In this paper, a method is proposed to solve the problem of monotone smoothing splines using general linear systems. This problem, also called monotone control theoretic splines, has been solved only when the curve generator is modeled by the second-order integrator, but not for other cases. The difficulty in the problem is that the monotonicity constraint should be satisfied over an interval w...

We present a construction of a re nable compactly supported vector of functions which is bi orthogonal to the vector of B splines of a given degree with multiple knots at the integers with prescribed multiplicity The construction is based on Hermite interpolatory subdivision schemes and on the relation between B splines and divided di erences The bi orthogonal vector of functions is shown to be...

Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) Fractional B-splines are a natural extension of classical B-splines. In this short paper, we show their relations to fractional divided differences and fractional difference operators , and present a generalized Hermite-Genochi formula. This formula then allows the definition of a larger class of fractional B-splines.

This essay reviews those basic facts about (univariate) B-splines that are of interest in CAGD. The intent is to give a self-contained and complete development of the material in as simple and direct a way as possible. For this reason, the B-splines are defined via the recurrence relations, thus avoiding the discussion of divided differences which the traditional definition of a B-spline as a d...

Triangular B-splines are powerful and flexible in modeling a broader class of geometric objects defined over arbitrary, non-rectangular domains. Despite their great potential and advantages in theory, practical techniques and computational tools with triangular B-splines are less-developed. This is mainly because users have to handle a large number of irregularly distributed control points over...

We extend Schoenberg’s family of polynomial splines with uniform knots to all fractional degrees α > −1. These splines, which involve linear combinations of the one-sided power functions x+ = max(0, x) α, are α-Hölder continuous for α > 0. We construct the corresponding B-splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains....