The multivariate splines which were first presented by deBooor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development...The multivariate splines which were first presented by deBooor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development.The author of this paper is interested in the area of inter- polation with special emphasis on the interpolation methods and their approximation orders. But such B-splines(both univariate and multivariate)do not interpolated directly,so I ap- proached this problem in another way which is to extend my interpolating spline of degree 2n-1 in univariate case(See[7])to multivariate case.I selected triangulated region which is inspired by other mathematicians'works(e.g.[2]and[3])and extend the interpolating polynomials from univariate to m-variate case(See[10])In this paper some results in the case m=2 are discussed and proved in more concrete details.Based on these polynomials,the interpolating splines(it is defined by me as piecewise polynomials in which the unknown par- tial derivatives are determined under certain continuous conditions)are also discussed.The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated.We lunited our discussion on the rectangular domain which is partitioned into equal right triangles.As to the case in which the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains,we will discuss in the next pa- per.展开更多
文摘The multivariate splines which were first presented by deBooor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development.The author of this paper is interested in the area of inter- polation with special emphasis on the interpolation methods and their approximation orders. But such B-splines(both univariate and multivariate)do not interpolated directly,so I ap- proached this problem in another way which is to extend my interpolating spline of degree 2n-1 in univariate case(See[7])to multivariate case.I selected triangulated region which is inspired by other mathematicians'works(e.g.[2]and[3])and extend the interpolating polynomials from univariate to m-variate case(See[10])In this paper some results in the case m=2 are discussed and proved in more concrete details.Based on these polynomials,the interpolating splines(it is defined by me as piecewise polynomials in which the unknown par- tial derivatives are determined under certain continuous conditions)are also discussed.The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated.We lunited our discussion on the rectangular domain which is partitioned into equal right triangles.As to the case in which the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains,we will discuss in the next pa- per.
