The layout of the survey is as follows. After a short discussion of cardinal B-splines, i.e., of B-splines on a uniform knot sequence, in Section 2, B-splines for an arbitrary knot sequence are introduced in Section 3 and shown to be a basis for certain spaces of piecewise polynomial functions. Various simple properties of B-splines are listed in Section 4, and the relationship between a spline...

A refinable spline in R is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best known refinable splines in R are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer aided geometric design. Such functions have be...

In this paper, quadratic and sextic B-splines are used to construct an approximating function based on the integral values instead of the function values at the knots. This process due to the type of used B-splines (fourth order or sixth order), called integro quadratic or sextic spline interpolation. After introducing the integro quartic and sextic B-spline interpolation, their convergence is ...

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...

Tchebycheffian splines are smooth piecewise functions whose pieces drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most the properties known in spline case. In particular, under suitable assumptions, admit representation terms basis functions, called B-splines (TB-splines), completely analogous to B-splines. A particularly ...

Most of the Soft Computing algorithms will learn from human knowledge and mimic human skills. We survey the principal constituents of soft computing techniques including Fuzzy Logic, Artificial Neural Networks, Support Vector Machines, Probabilistic Reasoning, Genetic Algorithms and Multi-Variate Adaptive Regressive Splines. Soft Computing techniques are being widely used by the IDS community d...

Recently, we have introduced spaces of splines deened on trian-gulations lying on the sphere or on sphere-like surfaces. These spaces arose out of a new kind of Bernstein-B ezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for such spline spaces analogous to the well-known theory of polynomial splines on planar triangulations. ...

A new global basis of B-splines is defined in the space of generalized quadratic splines (GQS) generated by Merrien subdivision algorithm. Then, refinement equations for these B-splines and the associated corner-cutting algorithm are given. Afterwards, several applications are presented. First a global construction of monotonic and/or convex generalized splines interpolating monotonic and/or co...

We bound the dimension of xed degree real algebraic interpolatory spline spaces. For a given planar triangulation T real algebraic splines interpolate speciied zi values at the vertices vi = (xi; yi) of T. For a three dimensional simplicial complex ST , real algebraic splines interpolate the boundary vertices vj = (xj; yj; zj) of ST. The main results of this paper are lower bounds on the dimens...

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local vari...