Uniform Beta-like Splines Of Order N

Beta-like spline functions are generated from one ECT system which is shifted by integer translation via one connection matrix. If the matrix is nonsingular, lower triangular and totally positive, there exists Beta-like splines having minimal compact supports. The Beta-like splines share most properties of polynomial B splines. In this paper, the uniform Beta-like splines are only discussed .From this a new kind of curves and surfaces are constructed. The research achievements and main contents are as follows:1. The uniform Beta-like spline space S (U , A, X ) is set up from the space (?) = spanE. The uniform Beta-like spline basis on the space S (U , A, X )is explicitly represented by extending the cardinal ECT B splines to the case for equal knots, from which some important properties of the basis are derived and the existence and uniqueness of the uniform interpolating functions on the space S (U , A, X ) are proved as well.2. By taking the different four pairs of functionsφ( x ), ? ( x), the uniform Beta-like splines above become the polynomial Beta splines, the algebraic trigonometric Beta splines, the algebraic hyperbolic Beta splines and the rational polynomial Beta splines for equal knots respectively, which again reduce to the polynomial B , the algebraic hyperbolic B and the rational polynomial B splines for equal nodes respectively when each connection matrix is identity matrix. Their explicit representations are given in the paper, in particular for the cases of the splines of low order.3. The generation of a kind of curves and tensor product surfaces by the new splines are given, and some their properties are shown in the paper.4. Two algorithms for the knot insertion into the uniform Beta-like spline curves are provided. One is based on recursive formulas,The other is from the generalized Pólya polynomials.