Interpolation by new B-splines on a four directional mesh of the plane

In this paper we construct new simple and composed B-splines on the uniform four directional mesh of the plane, in order to improve the approximation order of B-splines studied in Sablonnière (in: Program on Spline Functions and the Theory of Wavelets, Proceedings and Lecture Notes, Vol. 17, University of Montreal, 1998, pp. 67–78). If φ is such a simple B-spline, we first determine the space P(φ) of polynomials with maximal total degree included in S(φ)={∑α∈Z2c(α)φ(.−α), c(α)∈R}, and we prove some results concerning the linear independence of the family B(φ)={φ(.−α),α∈Z2}. Next, we show that the cardinal interpolation with φ is correct and we study in S(φ) a Lagrange interpolation problem. Finally, we define composed B-splines by repeated convolution of φ with the characteristic functions of a square or a lozenge, and we give some of their properties.