Study On Lagrange Interpolation Of Non Algebraic Surface

The multivariate interpolation theory cannot be directly obtained by the simple promotion of the univariate interpolation theory.Hence,the definition of the properly posed set of nodes for interpolation on a non-algebraic surface is given to study their properties and construction methods based on the basic algebraic theory and the theory of interpolation on an algebraic surface to address the problem of constructing multivariate interpolation nodes when a non-algebraic surface is given.In this way,the method is given for constructing the properly posed set of nodes for interpolation on a non-algebraic surface in addition to solving its existence problem on the non-algebraic surface with continuousG₀.Also,the construction of the obtained set of node is proven to be feasible and effective by citing the illustration of specific interpolation examples.The Lagrange interpolation problem on a non-algebraic surface is discussed in three chapters,as shown below.Chapter One consists of three sections.To be specific,the feasibility of multivariate polynomial interpolation is demonstrated by the Stone-Weierstrass theorem and its corollary in the first section.Then the second section introduces the formulation of multivariate interpolation.That is,the general multivariate interpolation formulation is given before explaining the multivariate classical Hermite interpolation problem as a special case.Eventually,it is proved that the multivariate Lagrange interpolation problem is a special case of the multivariate classical Hermite interpolation problem.What is discussed in the third section is the interpolation of n-degree polynomial space with k elements before giving the necessary and sufficient conditions for the point set to form its adaptive set of node.Chapter Two introduces the interpolation theory of algebraic curve surface with three sections.Specifically,the definition,necessary and sufficient conditions and construction methods of interpolation on an algebraic surface are discussed in the first section.And the corresponding theorem of the Bezout theorem with a similar effect in ternary is given and proved in the second section based on the above introduction.After that,the concept of sufficient intersection is introduced in the third section,with the given definition,necessary and sufficient conditions and construction method of interpolation along spatial algebraic curves.Chapter Three is divided into four sections to study the multivariate Lagrange interpolation problem on a non-algebraic surface.In the first section,the concept of the properly posed set of nodes for interpolation on a non-algebraic surface is given,with some properties proved that include the existential uniqueness of interpolation functions.The second section are presented with the concept of structure set,which also proves that there must be a properly posed set of nodes for interpolation on a non-algebraic surface with continuous ₀G with some lemma given.Based on this,the strict concept of a non-algebraic surface and the construction method of the properly posed set of nodes for interpolation on a non-algebraic surface are given and proved.Finally,specific examples are demonstrated in the fourth section.