On Multivariate Splines And Multivariate Continued Fractions Interpolation

During many years a vast amount of work has been done on the subject of multivariate approximation that is an increasingly active research area nowadays. Since much classical univariate theory does not straightforwardly generalize to the multivariate one, the field is both fascinating and intellectually stimulating. As a result, new tools have been devel-oped, such as multivariate splines, radial-basis functions, rational approximation and so on. Some problems in bivariate cubic spline functions and multivariate continued fractions interpolation over non-rectangular meshes are studied in this dissertation. We summary the contents as follows:In the first Chapter, we presents some preliminaries for the dissertation, including multivariate splines, the Conformality of Smoothing Cofactor Method, and the definitions, the properties and some basic facts of continued fractions.In the second Chapter, by using the dimension of multivariate spline space, we discuss the dimension of nonuniform bivariate spline space S31,2(â–³mn(2)), which is the same as that of the uniform case. Moreover, by means of the Conformality of Smoothing Cofactor Method, the basis of S31,2(â–³mn(2)) composed of two sets of splines are worked out in the form of the values at ten points in each triangular cell, both of which possess distinct local supports. Furthermore, we obtain the explicit coefficients in terms of B-net of the two sets of splines, respectively.In the third Chapter, by using the basis composed of two sets of splines with distinct local supports, we investigate cubic spline quasi-interpolating operators on nonuniform type-2 triangulation. These variation diminishing operators based on five mesh points or the center of the support of each spline Bij1 and five mesh points of the support of each spline Bij2 can preserve good approximation, and Vmn(f) even reproduce any cubic polynomial of nearly best degrees. Moreover, the spline series can approximate a real sufficiently smooth function uniformly based on the modulus of continuity. Furthermore, the derivatives of the nearly optimal variation diminishing operator can approximate that of the real sufficiently smooth function uniformly over quasi-uniform type-2 triangulation. And then we work out the convergence results. In the fourth Chapter, by using the barycentric coordinates expression of the interpo-lating polynomial over each ortho-triple, we obtain some properties. Moreover, we work out the explicit coefficients in terms of B-net for one ortho-triple, and two ortho-triples, respectively. Thus the computation of multiple integrals can be converted into the sum of the coefficients in terms of the B-net over triangular domain much effectively and con-veniently. Based on a new symmetrical algorithm of partial inverse differences, a novel continued fractions interpolation scheme is presented over arbitrary ortho-triples in R2, which is a bivariate osculatory interpolation formula with one-order partial derivatives at all corner points in the ortho-triples. Furthermore, we present its characterization theo-rem by using three-term recurrence relations. The new scheme is advantageous over the polynomial one with some numerical examples.In the last Chapter, we determine three-term recurrence relations for branched contin-ued fractions. Based on the algorithm of partial inverse differences in tensor-product-like manner, the finite branched continued fractions can be applied to rational interpolation over pyramid-typed grids in R3. By means of the three-term recurrence relations, a characterization theorem is valid. Then we work out an error estimation. By using the relationship between the partial inverse differences and partial reciprocal ones, and the partial reciprocal derivatives as well, we state the BCFs osculatory interpolation with its algorithm, which shows it feasibility of partial derivable functions in BCFs expansion at one point.