| Interpolation is a very classic problem of Mathematics and also a basic problem in computational Mathematics. It is well known that univariate interpolation has a very well developed theory and method. However, since 1980's , people come to turn the research of interpolation on multivariate interpolation, the reason is that multivariate interpolation has a widespread application in many fields, such as surfaces design, multivariate function arrange figures and finite element method.The most commonly method in multivariate interpolation is multivariate polynomial interpolation, that means for any given set of nodes, we seek a multivariate polynomial function which can represent any multivariate function approximately, but whose values at prescribed points (i.e., nodes) are coincidence. The primary problem when we are getting into multivariate polynomial interpolation is the existence of unique of interpolation polynomial function, which is our major working with and is known as properly posed for multivariate polynomial interpolation. Due to it has closely related to the scheme of interpolation, the properly posed of multivariate polynomial interpolation is become a very important aspect on the theory of multivariate interpolation and also a fashionable object of research recently. There exists two ways in this research:(1) to find the properly posed set of interpolation conditions for a given space of interpolating polynomials;(2) to find the properly posed space of interpolating polynomials for a given set of interpolation condition.Prof. C. De Boor,Ron, and T. Sauer have gotten some excellent results on the way (2), but our research work keeps straight on the way (1).In 1965,Prof.Liang [1]first posed a basic concept of the properly posed set2 Multivariate Lagrange Interpolation and Multivariate Kergin Interpolationof nodes (or PPSN,for short) for bivariate Lagrange interpolation and gave a kind of recursive method of constructing properly posed set of nodes for bivariate Lagrange interpolation. In 1998, Liang and Lit posed the following concept of bivariate Lagrange interpolation along an algebraic curve without multiple factors ( or AC WMF, for short )and gave a kind of recursive method of constructing PPSN for bivariate Lagrange interpolation along an ACWMF in[2].Let n be a nonnegative integer and en =1/2 -{n + l)(n + 2). Pn2 denotes the space of all bivariate polynomials of total degree < n, i.e.In 1965, Liang gave the following definition in [1] about properly posed set of nodes for bivariate Lagrange interpolation.Definition A (Liang,1965) Let A = {Qi}1=i be a set of en distinct points on E2. For any given set {fiYili of real numbers, we seek a polynomial p(x,y) G P? satisfying :p{Qi) = fi, i = l,---,en. (1)// for any given set {fiY=i °f rea^ numbers there exists a unique solution for the equation system (1), we call the interpolation problem a properly posed interpolation problem, and call the corresponding set A = {Qi)1=i of nodes a PPSN for Pi2).Theorem A(Liang,1965) Suppose A = {Qi}Zi ? a PPSN for P^2) and none of these points lies on an irreducible algrbraic curve q(x,y) = 0 of degree k (k = 1, 2; k = 1 means q(x,y) = 0 is a straight line; k = 2 means q(x,y) = 0 is a conic ), then A = {Qi}1=i together with the (n + 3)k — 1 distinct points selected freely on the irreducible algrbiaic curve q(x,y) = 0 of degree k must constitute a PPSNforF%k.Since any point in E2 must constitute a PPSN for Pq , basing on it, we can construct PPSN for a series of interpolation spaces by using Theorem A repeatedly. At the same time we call the constructive methods given in TheoremA Line-Superposition Process ( k = 1) and Conic-Superposition Process (k = 2) respectively.Definition B(Liang and Lit,1998) Let k be a natural number, and q(x, y) = 0 be an AC WMF of degree k,n + 2 n + 2-fe ri(n+l)(n + 2)> n-k), n>k.Also suppose B = {Qi}il\ is a set of the en(k) distinct points on curve q(x,y) = 0. For any given set {/i}^ of real numbers, we seek a polynomial p(x,y) G Fn satisfying = fi, i=h---,en(k). (2)If for each given set {/i}^ of real numbers, there always exists a solution for the equation system (2), we call the set B = {Qi}il\ of nodes a PPSN for polynomial interpolation of degree n along the curve q(x,y) = 0 of degree k, and write B G In (q) ( where In \q) denotes the set of all PPSN for polynomial interpolation of degree n along the ACWMF q(x,y) = 0 ).Theorem B(Liang and Lit,1998) Suppose an ACWMF q(x, y) = 0 of degree k and a straight line l(x,y) = 0 meet exactly at k distinct points C = {Qi}i=1, B G In2\q) (n>k-2),andBC\C = 0. Then we haveB\JC G I^iq).We call the constructive method given in Theorem B a Line-Superposition Process of constructing PPSN for polynomial interpolation along an algebraic curve.Theorem C(Liang and Lit,1998) Let the set A = {Qi}t=i of points be a PPSN for Pn . // none of these points lies on an ACWMF q(x,y) = 0 of degree k, then for any V e In2lk(q), VUA must be a PPSN for pJJ^,.We call the constructive methods given in Theorem C an Algebraic Curve-Superposition Process of constructing PPSN for bivariate Lagrange interpolation . According to theorem B and theorem C, we know that the constructive methods4 Multivariate Lagrange Interpolation and Multivariate Kergin Interpolationgiven in Theorem A Line-Superposition Process and Conic-Superposition Process is a special cases when take k = 1 and k = 2 in theorem C.Liang and Lit researched Lagrange interpolation along an algebraic surface and acquired the result as follow:Theorem D(Liang and Lit,2001) Suppose an algebraic surface without multiple factors q(X) = 0 of degree k and a plane h(X) = 0 meet at a plane algebraic curve C = s(q, h).If A = {Qi}i=i €= In (q)and no one point in A contained in the curve C = s(q,h),B E I^C) [n>k- 2).ThenA\JB E I^q).We call the constructive method given in Theorem D Plane-Superposition Process of constructing PPSN for polynomial interpolation along an algebraic surface.On the other hand ,Liang and Lit went deep into studying [22,75]the convergence of Kergin interpolation polynomial and acquired [24] the results of uniform convergence and integral convergence of Kergin interpolation and its derivariate for the smooth functions defined on the unit disk in 1998.Our research in this paper is the continuation and deepen of the previous works.The main results in the paper as follow:1 . We go a step further research the problem of PPSN for Lagrange interpolation along a plane algebraic curve .We introduce a concept of weak Grobner basis and by using the conclusion of Cayley-Bacharach theorem in algebraic geometry, we give a generalize method for constructing PPSN along an ACWMF and get some deepen results on PPSN for bivariate Lagrange interpolation, expand the Theorem B .At same time,we acquire some corollaries which are convenient to be used and make clear the geometric construction and the basic characteristric on bivariate Lagrange interpolation.Definition l(weak Grobner basis) Suppose Pi E K[x\,--- ,xn] for i = 1, ? ? ? ,m, degpi = U and I =< pi, ? ? ? ,pm >■If for each polynomial p G I f]Fns', we can always seek polynomials ai E K[x\,- ■■, xn], i = 1, ? ? ? , m such that p = Y^iLi ai'Pi and degai < n — li; i = 1, ? ? ? ,m, then we call the set of polynomials {pi, ■■■,Pm} is a weak Grobner basis for I.Theorem 1 Suppose an ACWMF q(x,y) = 0 of degree k and an algebraic curve p(x,y) = 0 of degree 1(1 > 1) meet exactly at Ik distinct points C = {QiYiLi,{p, q} is a weak Grobner basis for < p,q >, B E In (q) (where n > k — 2) and BC\C = 0. Then we haveTheorem 2 Suppose an ACWMF q(x,y) = 0 of degree k and a conic p(x,y) = 0 meet exactly at 2k distinct points C = {Q^I^.IfB G In (q)(where n > k — 2) and B (\C = 0. Then we haveBuCeIn%(q).Proposition 1 Suppose an algebraic curve q(x,y) = 0 of degree k and a conic p(x,y) = 0 meet exactly at 2k distinct points {Qi}f=i - Then {p,q} must be a weak Grobner basis for < p,q >.In the paper,by using the conclusion of Cayley-Bacharach theorem in algebraic geometry ,we get some deepen results and practical corollaries on PPSN along a plane algebraic curve.Theorem 3 Let {0} = P^J = P^ = P^..., denote the space of zero polynomials, and under these circumstances we regard their corresponding PPSN as an empty set. Suppose m, n are natural numbers, m < n, and a is a integer number satisfying a > 1 — m. Also suppose that a curve p(x,y) = 0 of degree m and another q[x,y) = 0 of degree n meet exactly at mn distinct points A = {Qi}^,and B C A is a PPSN for P^; then we have(1) IfC\ is a PPSN for polynomial interpolation of degree rn + a — 3 along the curve p(x, y) = 0 of degree m, and C\f]A=0, then C\ U(^-\^) must be a PPSN for polynomial interpolation of degree m + n + a — 3 along the curve p(x, y) = 0 of degree m ;(2) IfC2 is a PPSN for polynomial interpolation of degree n + a — 3 along the curve q(x, y) = 0 of degree n, and C2f]A=0, then C2 \J(A\B) must be a PPSN for polynomial interpolation of degree m + n + a — 3 along the curve q(x, y) = 0 of degree n.From Theorem 3, we deduce three corollaries which are convenient to be used as follows.6 Multivariate Lagrange Interpolation and Multivariate Kergin InterpolationCorollary 1 Suppose m and n are natural numbers, m < n, r is nonnegative integer, and 3 < r < m + 2. // the curve p(x, y) = 0 of degree m and q(x, y) = 0 of degree n meet exactly at mn distinct points A = {Qi}1?^, and the set B C A is the PPSN for P^3, then(1) If T>i is a PPSN for polynomial interpolation of degree m — r along the curve p(x,y) = 0 of degree m , T>i f]A = 0, then V>\ \J(A\B) must constitute a PPSN for polynomial interpolation of degree m + n — r along the curve p(x,y) = 0(2) If T>2 is a PPSN for polynomial interpolation of degree n — r along the curve q(x,y) = 0 of degree n , V2 f]A = 0, then V2 \J(A\B) must constitute a PPSN for polynomial interpolation of degree m + n — r along the curve q(x,y) = 0Corollary 2 Suppose m and n are natural numbers, and r is nonnegative integer. If the curve p{x,y) = 0 of degree m and q{x,y) = 0 of degree n meet exactly at mn distinct points A = {Qi}?", then(1) If S\ is a PPSN for polynomial interpolation of degree m + r — 2 along the curve p(x, y) = 0 of degree m, ￡if)A=0, then Si\JA must constitute a PPSN for polynomial interpolation of degree m + n + r — 2 along the curve p(x, y) = 0 ;(2) If ￡2 is a PPSN for polynomial interpolation of degree n + r — 2 along the curve q(x, y) = 0 of degree n , ￡2 f] A = 0, then ￡2 \J A must constitute a PPSN for polynomial interpolation of degree m + n + r — 2 along the curve q(x, y) = 0 .Corollary 3 Suppose m and n are natural numbers, and r is nonnegative integer. If curve p{x, y) = 0 of degree m and q(x, y) = 0 of degree n meet exactly at mn distinct points A = {Qi}?", then(1) If Bx e iP (p) , andBlf]A=0, then Bl \J A e I^n(p);(2) If B2 G /i2)(q) ,andB2^A= 0, then B2[}Ae lflm{q).From the Corollary 3, it is obvious that if we take n = 1 in the Corollary 3 (1) and take m = 1 in the Corollary 3(2) respectively,then theorem C can be acquired.That is to say, theorem B is a specalic case of the Corollary 3.2. We propose the concepts which sufficient intersection of algebraic surfaces and Lagrange interpolation along a space algebraic curve and give the sufficient and necessary condition for a set of nodes lie on a space algebraic curve canbe constitute a PPSN for along this curve and get a most generally method of constructing PPSN for Lagrange interpolation along a space algebraic curve,that is an Algebraic Surface Postition Process for constituting PPSN along an algebraic surface.The research on properly posed set of nodes for Lagrange interpolation along a space algebraic curve and an algebraic surface(for example, sphere) have great practical applied value. They are expensively used to explore in atmosphere,marine resources even though in the reconstruction of blood vessel and nervous net.Definition 2 Let I and k be natural numbers, we call an algebraic surface p(X) = 0 of degree I and another q(X) = 0 of degree k sufficient intersection in a space algebraic curve C = s(p,q), if there exists a plane h(X) = 0 such that it and the space algebraic curve C = s(p,q) meet exactly at Ik distinct points .Definition 3 Let n, I and k be natural numbers and en(l,k) be defined as follows: ( + 3\ f n-l + 3\ (n-k + 3\ (n-l-k + ?>\.. 3 )-( 3 H 3 H 3 j(3)Suppose an algebraic surface p(X) = 0 of degree I and another q(X) = 0 of degree k sufficient intersection in a space algebraic curve C = s(p,q), A = {QiYi=i *s a set °fen(l,k) distinct points on the curve C = s(p,q). If for each given set {fiYi=i °f rea^ numbers, there always exists a polynomial g{X) G fn satisfyingg(Qi) = fi,i=l,--- ,en(l,k),then we call the set A = {QiYli of nodes a PPSN for Lagrange interpolation along the space algebraic curve C = s(p,q). In this case, we write A = {Qi}Zfk) e In\C)( where I{n\C) denotes the set of all PPSN for Lagrange interpolation along the space curve C = s(p,q)J.Theorem 4 Let m, n and k be nonnegative integer numbers, an algebraic surface q(X) = 0 of degree k and an algebraic surface p(X) = 0 of degree m intersects sufficiently in a space algebraic curve C = s(p,q). An algebraic surface r(x) = 0 of degree I meets curve C = s(p,q) exactly at mkl distinct points A = {Qi\?=l, tf&￡ In\C)(n >m + k-3) and B f]A = 0, then we haveMultivariate Lagrange Interpolation and Multivariate Kergin InterpolationTheorem 5 Let m, n and k be nonnegative integer numbers, an algebraic surface p(X) = 0 of degree k and an algebraic surface q(X) = 0 of degree m intersects sufficiently in a space algebraic curve C = s(p,q). We take a set A ￡ In (q) and no any point in A lies on the carve C = s(p, q). If B G In+m(C), then we haveIt is obvious that Theorem 5 expands Liang's result in 2001 [5],i.e.,the theorem D which is about a Plane-Superposition Process for constructing a PPSN along an algebraic surface to the case an Algebraic Surface-Superposition Process.3. In the paper, we give a new proof for Cayley-Bacharach theorem in algebraic geometry by means of interpolation and extend the theorem which valid only in E2 to the case is holds in E3.At same time, by using the conclusion of Cayley-Bacharach theorem holds in E3, we get some practical methods which constructing PPSN for Lagrange interpolation along a space algebraic curve and get some deeped results for Lagrange interpolation along a space algebraic curveTheorem 6 Let {0} = pf3} = P^ = P^..., denote the space of zero polynomials, and under these circumstances we regard their corresponding PPSN as an empty set. Suppose m, n and k are natural numbers, m < n < k, and a is a integer number satisfying a > 1 — m. Also suppose that algebraic surfaces p(X) = 0 of degree m , q(X) = 0 of degree n and r(X) = 0 of degree k meet exactly at rank distinct points A = {Qi}^,and B C A is a PPSN for P^i, then we have(1) If A\ is a PPSN for polynomial interpolation of degree m + n + a — 4 along the space algebraic curve C\ = s(p, q) , and A\ P| A = 0, then A\ |J(^4.\￡>) must be a PPSN for polynomial interpolation of degree m + n + k + a — 4 alongthe curve C\ = s(p,q) ;(2) If A2 is a PPSN for polynomial interpolation of degree n + k + a — 4 along the space algebraic curve C2 = s(q, r), and A2 f] A = 0, then A2 U(A\B) must be a PPSN for polynomial interpolation of degree m+n+k+a— 4 along the curve C2 = s(q,r);(3) If A3 is a PPSN for polynomial interpolation of degree m + k + a — 4 along the space algebraic curve C3 = s(p, r) ; and A3 f] A = 0, then A3 \J(A\B) must be a PPSN for polynomial interpolation of degree m + n + k + a — 4 along the curve C3 = s(p,r).From Theorem 8, we deduce three corollaries which are convenient to be used as follows.Corollary 4 Suppose m , n and k are natural numbers, m < n < k, 4:i is a PPSN for polynomial interpolation of degree rn+n — r along the space algebraic curve C\ = s(p,q), T>if)A= 0, then V>\ \J(A\B) must constitute a PPSN for polynomial interpolation of degree m + n + k — r along the curve G\ = s(p, q) ;(2) If V>2 is a PPSN for polynomial interpolation of degree n + k — r along the space algebraic curve C2 = s(q,r) , T>2f)A = 0, then T>2\J(A\B) must constitute a PPSN for polynomial interpolation of degree m + n + k — r along the curve C2 = s(q, r) ;(3) IfT>3 is a PPSN for polynomial interpolation of degree m + k — r along the space algebraic curve C3 = s(p,r) , V3 f] A = 0, then T>3\J(A\B) must constitute a PPSN for polynomial interpolation of degree m + n + k — r along the curve C3 = s(p, r) .Corollary 5 Suppose m , n and k are natural numbers, r is nonnegative integer. If algebraic surfaces p(X) = 0 of degree m , q(X) = 0 of degree n and r(X) = 0 of degree k meet exactly at mnk distinct points A = {Qi}?" , then(1) If S\ is a PPSN for polynomial interpolation of degree m + n + r — 3 along10 Multivariate Lagrange Interpolation and Multivariate Kergin Interpolationthe space algebraic curve C\ = s(p, q), ￡i[\A = 0, then ￡\{JA must constitute a PPSN for polynomial interpolation of degree m + n + k + r — 3 along the curve C\ = s(p, q) ;(2) If ￡2 is a PPSN for polynomial interpolation of degree n + k + r — 3 along the space algebraic curve C2 = s(q,r), ￡2^ A = 0, then ￡2 \J A must constitute a PPSN for polynomial interpolation of degree m + n + k + r — 3 along the curve(3) If ￡3 is a PPSN for polynomial interpolation of degree m + k + r — 3 along the space algebraic curve C3 = s(p, r), ￡3 f] A = 0, then ￡3 (J A must constitute a PPSN for polynomial interpolation of degree m + n + k + r — 3 along the curve C3 = s(p,r) .Corollary 6 Suppose m , n and k are natural numbers, r is nonnegative integer. If algebraic surfaces p(X) = 0 of degree m , q(X) = 0 of degree n and another r(X) = 0 of degree k meet exactly at rank distinct points A = {Qi}?" , then(1) IfTx G lf\Cx = s(p,q)) , andTx{\A= 0, then Tx \^A e ll%(C\ = s(p,q));(2) If T2 e iP (C2 = s(q, r)) , and T2 f| A = 0, then T2 U A G ifljC* = s(q,r));(3) If 3=3 G lf> (C3 = s(p, r)) , and T3 f| A = 0, then T3 U A G /^(C3 = s(p,r)).The conctens as above part 2 and part 3 are original and all news results .4. We discuss the weak convergence of Kergin interpolation and proof the weighted integral convergence and weighted (partial) derivatives integral convergence of Kergin interpolation polynomial for the smooth functions defined on the unit disk. Those expands Liang's results in [24].In 1980, P.Kergin[18] proposed a new kind of multivariate interpolation, which is known as Keigin interpolation. In the same year, C.A.Micchelli[19] gave a Newton representation of it and based on this representation, he established the error estimate of Kergin interpolation. Also using this representation, L.Bos (1983) and X.Z.Liang(1986) researched the convergence of Kergin interpolation for the analytic functions in the unit disk(see [21,75]). In 1989,X.Z.Liang[22] gavea Lagrange representation of Kergin interpolation , based on this representation, Liang and Lit discussed in 1990 and 1998 the uniform convergence and the integral convergence respectly of Kergin interpolation and its derivatives for the smooth functions on the unit disk(see [23,24]). The aim of this paper is to research the weighted integral convergence of Kergin interpolation and its derivatives for the smooth functions on the disk.Let D denote unit disk in real plane IR2,i.e.,D = {X= (x,y)TmR2 | x2 + y2 < 1},Cl(D) denotes the space of all /-times continuously differentiable functions defined in D , also n > 1 be natural number. Set0i = —,xi =njk = nkj = (cos-(9j + 9k), sin-(9j + 9k))T, 1 < j < k < n.For f(x) E C°(D), X, Y e E2, we introduce integral functional as follows:/= r f(XX+(l-X)Y)d\,[X,Y] JOand letTm(t) = cos(marccost), Um{t) = T'm{t),= rm(t)+rm-!(t) = rm(t)-rmmU t+1 , mU t1X = (x,y)T = (rcos9,rsin9)T,U = rcos(-----9) = cos—x + sin—y, i = 0,1, 2, ? ? ?n n nOn the basis of the results in [22], Liang give the following Theorem in [23].Theorem E(Liang,1990) For any f(X) E C1(D), there exists a unique polynomial Kn(X) = Kn(f;X) E P^i, such that the following conditions are satisfied.X') = f(X'), l 0, 0 < a < 1), then for any f(X) G Cl(D), we have^ f(X)W(X)dX.D Dd2 d2Theorem 8 Suppose W{X) = W(x,y) G C2(D), —W(x,y),—W(x,y) G>}dx2 LipaM(M > 0, 0 < a < 1) then for any f(X) G C2(D), we haveJJ ^Kn{X)W{X)dX = JJ j-f(X)W{X)dX-D DD DTheorem 9 For any natural number n > 2,X G D and f(X) G Cl(D),then we have:Jj(Kn(f;X)-f(X))W(X)dXD...
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