| 摘 要： | The multivariate splines which were first presented by deBooor as a complete theoreticalsystem have intrigued many mathematicians who have devoted many works in this field whichis 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 degree2n-1 in univariate case(See7])to multivariate case.I selected triangulated region which isinspired by other mathematicians'works(e.g.2]and3])and extend the interpolatingpolynomials from univariate to m-variate case(See10])In this paper some results in thecase m=2 are discussed and proved in more concrete details.Based on these polynomials,theinterpolating 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.Theapproximation orders of interpolating polynomials and of cubic interpolating splines areinverstigated.We lunited our discussion on the rectangular domain which is partitioned intoequal right triangles.As to the case in which the rectangular domain is partitioned into unequalright triangles as well as the case of more complicated domains,we will discuss in the next pa-per.
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