Lagrange Interpolation In N Variables And Wavelet Approximation For Two-dimensional Digital Image

Lagrange interpolation in n variables and wavelet approximation for two-dimensional digital image are two hot problems for discussion in the field of multivariate approximation. In this pater, we just research in these two problems and obtain some corresponding results. Part 1. The research of Lagrange interpolation in n variablesPolynomial interpolation at all times is a significant aspect for study in computational Mathematics. Specially when it was widely applied in many practical problems, such as multivariate function arrange figures, surface design and finite element method and so on, the theory and method for multivariate polynomial interpolation was rapidly developed in recent twenty and thirty years.It is well known that the theory and method for univariate interpolation has been developed perfectly. But since multivariate polynomial interpolation is defined in n-dimensional space, so it has more complex characters than those of univariate interpolation. In univariate polynomial interpolation the monomials are arranged in a unique order and the number of roots is uniquely confirmed by the degree of polynomial. But multivariate polynomial interpolation has not the above characters, so that the multivariate polynomial for interpolation can not be uniquely confirmed even when the number of nodes for interpolation is equal to the dimension of interpolation space. So at first we need to resolve the problem about the properly interpolating. At present there are two main ways to research this problem: one way is to find out a proper space of polynomial interpolation for a given set of interpolation nodes, specially to determine the minimal degree interpolation space.Some better results have been obtain by C. de Boor, T.Sauer and Y.Xu etc.; the other way is to construct the properly posed set of nodes for given space of interpolation. The latter is the main interested aspect for our paper.In 1965, Prof. Liang [9] firstly posed a basic concept of properly posed set of nodes (or PPSN, for short) for bivariate Lagrange interpolation and at the same time gave a kind2 Lagrange Interpolation in n variables and Wavelet Approximation for 2-D Digital Imageof recursive method of constructing PPSN for Lagrange interpolation in R2 namely Line-Superposition Process and Gonic-Superposition Process. In 1998, Liang and Lii[10] further posed the concept of PPSN along an algebraic curve without multiple factors (or ACWMF, for short) and gave the method of constructing PPSN along ACWMF by the intersection between a line and an algebraic curve of degree k. In 2003, Liang and Cui[24] researched Lagrange interpolation in M3. They posed the concepts of sufficient intersection of algebraic surfaces and PPSN for Lagrange interpolation along an algebraic surface and along a space algebraic curve, and deduced a general method of constructing PPSN along a space algebraic curve. They also extended Cayley-Bacharach theorem in algebraic geometry from M2 to R3.Our research is the continuation and advance of the previous work. In this paper we research deeply in the Lagrange interpolation along an algebraic hypersurface and an algebraic manifold in n-dimensional space. Our main results are as follows:Pm(n) denotes the space of all polynomials in n variables of degree < m with complex coefficients, i.e.We note the number of monomials in n variables of degree < m as M(n,m), thenThat is the dimension of the space of polynomials Definition 1 Let A = be a set of M(n,m) distinct points in Cn. Given any set of complex numbers, we seek a polynomial xn)) satisfying the interpolation conditions for any given set of complex numbers there always exist a unique solution for the equation system (2), we call the interpolation problem a properly posed interpolation problem and call the corresponding set A = of nodes a properly posed set of nodes (or PPSN, for short) for Theorem 1 [9] A set A = of points in C is a PPSN for if and only if is not contained in any algebraic hypersurface of degree m.Definition 2 Suppose](s is...