| Polynomial interpolation is an important subject of computational mathematics.The study of interpolation with multivariate polynomial has developed rapidly in the recent twenty years.In the research of multivariate interpolation, the well-posedness problem is the first problem. Thus there are two ways for the research of multivariate interpolation: one way is to construct the properly posed set of nodes (or PPSN, for short) for a given space of interpolating polynomials ( see [1]-[7]). The other way is to find out the proper space of interpolating polynomials for a given set of interpolation nodes, specially to determine the interpolation space of minimal degree ( see de Boor and A.Ron [8],T.Sauer[9][10]). The former is the main research aspect of this paper.In 1948, J.Radon[1] gave the Straight Line-Superposition Process for constructing the PPSN for bivariate polynomial space. Then Liang[2] deduced the Superposition Interpolation Process for constructing the PPSN for R2( including Conic-Superposition Process). They changed the well-posedness problem into a geometrical problem so that we can use algebraic and geometrical theories to study multivariate polynomial problem.In 1998, Liang and Lii[12] posed the concept of PPSN along an algebraic curve without multiple factors and gave the method of constructing PPSN along it by the intersection between a line and an algebraic curve of degree k.In 2003, Liang and Cui [13] researched Lagrange interpolation in K3.They posed the concepts of sufficient intersection of algebraic surfaces and PPSN for Lagrange inerpolation along an algebraic surface and along a space algrbraic curve, and deduced a general method of constructing PPSN along a space algebraic curve.In 2004, Liang and Zhang[14] researched Lagrange interpolation along the sufficiently intersected algebraic manifold in Rn.They proved the existence of the PPSN of arbitrary degree along sufficiently intersected algebraic manifold and deduced a general method of constructing PPSN along the algebraic manifold, that is, Superposition Interpolation Process.Our research is the continuation and impovement of the previous work. In this paper we research depply the Hermite interpoaltion along an algrbraic manifold in n-dimensional space. First, we give the description of the Hermite interpolation problem. Then we prove the Superposition Interpolation Process for Hermite interpolation along the algebraic manifold. They includ the results of Lagrange interpolation as particular cases so we regard our research in this paper is a continuation and impovement for the research of multivariate polynomial interpolation.Part 1.Hermite interpolation along the algebraic manifold.In this paper we manily consider polynomial interpolation in real space and we denote Pm(Rn) the real space of all n-variate polynomials of total degree≤m,i.e.where |α|=α1+…+αn,α1,…,αn denotes nonnegative integer.Before giving the Hermite interpolation problem, we introduce some notations.Let Am-{Qi:Qi=(xi,1,…,xi,n,1≤i≤M) denote a set of M mutually distinct points in Rn.Let Zn denotes the set of integral points in Rn andLetÏ„i=(Ï„i,1,…,Ï„i,n) denote n unit vectors of linearly independent andλi=λ(Qi) denote a positive integer corresponding to the point Qi,whereÏ„i,1,…,Ï„i,n.And letdenote the set of index corresponding to the point Qi.For each Ki,j∈I(Qi) we define its order by O(Ki,j)=(?)ki,j,ι.For Ki,j and Ki,v in I(Qi),we draw a directed line which parallels to axis between them and stipulate the descendent direction is pointing to lower order from higher order if the distance between them is 1.Thus we get a oriented graph of I(Qi) and denote it by g(Qi).Joining the directed segments of length 1 according to the descendent direction, we can get a descendent path. And the length of the path is the number of the directed segments of length 1 which joint the descendent path.If for each point Ki,j∈I(Qi) one can find a descendent path in g(Qi) whose length is O(Ki,j) and takes origin as its destination, then we call I(Qi) a tree indexing set. We define the grade of I(Qi) by G(Qi)=(?){O(Ki,j)}.Let I(Qi) be a tree indexing set and for each Ki,j∈I(Qi) we define the set of downstream ponits by B(Ki,j) which includ all ponits in the descendent paths from Ki,j to origin (not includ the point of Kij).For each point Qi we have the assumption that the conrresponding I(Qi) is a tree indexing set. And we sort the order of the indexing set {Ki,j} according to the following rules.Rule 1.1). If O(Ki,j)i,u),then Ki,ji,u;2). If O(Ki,j)=O(Ki,u),then sort the order of {Ki,j} according to its components,that is, if there hasι,0≤ι≤n-1 such thatthen Ki,ji,u.So we can get a order of the indexing set {Ki,j},that is,Because the descendent paths in the oriented graph of I(Qi) have origin as their's destinations we get Ki.0=(0,0,…,0).And we suppose Ki,j sort its order from the higher order to the lower order.We consider the following set of interpolating functionals: whereandαu(i,j)∈R.Now we give the description of this kind of Hermite interpolation problem in the space Pm(Rn).Problem 1.We consider the set of interpolating functionals just as (1) where(?)λ(Qi)=em(n).For any given set of real numbers (?),we want to find pn(X)∈Pm(Rn) such thatDefinition 1 If there always exists a unique polynomial pn(X)∈Pm(Rn) such that (2) is satisfied for any given real numbers (?),then we call Problem 1 well posed and call the set of interpolating functionals in (1) a properly posed set of interpolating functionals of degree m for Hermite interpolation in the space Pm(Rn) and Am= {Qi,1≤i≤M} is called the corresponding set of nodes.In some practical problem people often need to consider the interpolation along the algebraic manifold. So we give the Hermite interpolation problem along the sufficiently intersected algebraic manifold. We suppose s(1≤s≤n) algebraic hypersur-faces without multiple factors f1(X)=0,…,fs(X)=0 of degree k1,…,ks respectively,sufficiently intersect at the algebraic manifold Vn-s=v(f1,…,fs) in Rn.Let Um(Vn-s)={Qi,1≤i≤M} be a set of M mutually distinct points on the algebraic manifold Vn-s.The definition of the grade of Um(Vn-s) is G(Um)=(?){G(Qi)}.Now we give the description of this kind of Hermite interpolation problem along the algebraic manifold Vn-s.Problem 2.We consider the set of interpolating functionals just as (1) and(?)λ(Qi)=em(n)(k1,…,ks).For any given set of real numbers (?),we want tofind pn(X)∈Pm(Rn) such that Definition 2 If there always exists a polynomial pm(X)∈Pm(Rn) such that (3)is satisfied for any given set of real numbers (?),then we call Problem 2 well posed and call the set of interpolating functionals in (1) a properly posed set of interpolating functionab of degree m for Hermite interpolation along the algebraic manifold Vn-s and Um(Vn-s) is called the corresponding set of nodes.The following theorem is the corresponding superposition interpolation process for this kind of Hermite interpolation along an algebraic manifold.Theorem 1 Let f1(X)=0,…,fs(X)=0 be s(1≤s≤n) be algebraic hypersur-faces without multiple factors of degree k1,…,ks,respectively, and they sufficiently intersect at an algebraic manifold Vn-s=v(f1,…,fs) in Rn.Let L be a PPSIF of degree m+rks for Hermite interpolation along the algebraic manifold Vn-s and Um+rks(Vn-s) be the corresponding set of nodes where r is the grade of Um+rks(Vn-s). Let (?) be a PPSIF of degree m for Hermite interpolation along the algebraic manifold Vn-s+1-v(f1,…,fs-1) and Um(Vn-s+1) be the corresponding set of nodes. Suppose fs(X)=0 does not pass through any point in Um(Vn-s+1).Then L∪(?) must be a properly posed set of interpolating functionals of degree m + rk along the algebraic manifold Vn-s+1 and Um+rks(Vn-s)∪Um(Vn-s+1) be the corresponding set of nodes.Using this theorem we can get a series of properly posed set of interpolating functionals of higher degree from those of lower degree along the algebraic manifold.We deeply research a kind of algebraic manifold in R3,that is the sphere. We study Lagrange interpolation and Hermite interpolation on the unit sphere, that is to say, we make a concrete practice for the superposition interpolation process for Hermite interpolation along the algebraic manifold. And we get some schemes for interpolation on the sphere which include the corresponding results in the previous conferences.Part 2. Lagrange interpolation on the unit sphere.In this section, we research Lagrange interpolation on the unit spherein the polynomial space of degree 3Î n3={(?)αi,j,kxiyjzk,αi,j,k∈R}.LetÎ n(S2) denote the space of spherical polynomials of degree n, that is, the restriction of polynomials degree n in three variables to S2.It is well known that dimÎ n3=(?),dimÎ n(S2)=(n+1)2.The following is the definition of Lagrange interpolation problem on the unit sphere. Definition 3 Let n be a natural number and suppose a set of points Un={Qi}i=1(n+1)2 include (n+1)2 mutually distinct points on the unit sphere S2.If there exists a polynomial pn(X)∈Πn3 such thatfor any given set of real numbers {fi}i=1(n+1)2.Then we call the set of points Un a PPSN of degree n on S2.For Lagrange interpolation on the unit sphere, it is more convenient to work with spherical coordinates:It is corresponding to a latitude on the sphere for any given latitudinal angleφ(0≤φ≤π).Letλ≤n+1 be a positive integer and we choose arbitrarily a set consisting ofλlatitudinal angles and denote it byΦλ={φι|1≤ι≤λ,0≤φ1<φ2<…<φλ≤π},moreover, we denote theλlatitudes corresponding to theλlatitudinal angles by the set of latitudes L(Φλ).After computation, we obtain the dimension of the space spanned by the set of latitudes L(Φλ),that is,λ(2n+2-λ). Now we give a description of the interpolation problem along the set of latitudesL(Φλ).Problem 3: Let n and j be nonnegative integers. Supposeιandλ≤n+1 are positive integers and a set of latitudes L(Φλ) correspond toλlatitudinal angles 0≤φ1<φ2<…<φλ≤πon the unit sphere. Suppose {Qι,j:,0≤j≤2n+1-λ,1≤ι≤λ} denotes the set of mutually distinct points on the latitude z=cosφι,1≤ι≤λ.We want to seek a polynomial pn(X)∈Πn3 such that Definition 4 If there always exists a polynomial pn(X)∈Πn3 such that (4) is true for any given real numbers {fι,j},then we call that Problem 3 is well-posed and the set of points a properly posed set of nodes of degree n along the set of latitudes L(Φλ).Especially, we need to impose an additional symmetry whenλis an even number such asλ=2h where h andιare positive integers, that is,φ2h+1-ι=Ï€-φ1,φι∈(0,(?)),1≤ι≤h.We denote the set of longitudinal angles byΘαn,λ(α∈[0,2)) as following sinceλis a positive integer:1.Ifλ=1,then2.Ifλ>2 thenThen we choose the set of points Unλalong the A latitudes as following:1. Ifλis a odd number, thenUnλ={Qι,j|Qι,j=(sinφιsinθj,sinφιcosθj,cosφι),θj∈Θαn,λ,0≤j≤2n+1-λ,1≤ι≤λ}2. If A is an even number,thenUnλ={Qι,j|Qι,j=(sinφιsinθj,sinφιcosθj,cosφι),θj∈Θ0n,λ,0≤j≤2n+1-λ,1≤ι≤h}∪{Qι,j|Qι,j=(sinφιsinθj,sinφιcosθj,cosφι),θj∈Θ1n,λ,0≤j≤2n+1-λ,h+1≤ι≤λ}Now we give the result of the interpolation problem along the set of latitudes L(Φλ).Theorem 2 Let n be a nonnegative integer and supposeλ≤n+1 is a positive integer. Then the set of points Unλgiven as above is a properly posed set of nodes of degree n for Lagrange interpolation along the set of latitudes L(Φλ) in the spaceÎ n3.The following theorem is the reification of the superposition process for interpolation on the sphere.Theorem 3 Let Vn={Qi}i=1(n+1)2 be a properly posed set of nodes of degree n of the spaceÎ n(S2) and suppose a set of latitudes L(Φλ) consisting ofλlatitudes does not pass through any point in Vn.Let Un+λ={Qi}i=1(λ(2n+2+λ) be a properly posed set ofnodes of degree n+λalong L(Φλ).Then Vn∪Un+λ must be a properly posed set of nodes of degree n+λon the sphere S2.Moreover, for n-2,Theorem 3 contains 8 sets of 16 points being a properlyposed set of nodes on the sphere, n = 4 gives 16 sets of 25 points,…….In general,the number of differently properly posed set of nodes for each degree is a powder series, that is, 2n.Proposition 1 Let Mn denote the number of properly posed set of nodes in Theorem 3.Then Mn=2n.We give some schemes for Lagrange interpoltation on the sphere and numerical examples and a algorithm to compute the polynomial for interpolation on the sphere.We will use the following transform of rotation axes to obtain an extension of our work in this paper.whereθ0∈[0,2Ï€],φ∈[0,Ï€].We rotate the set of latitudes L(Φλ) by means of above transform and get a set of parallel circumferences (?)(Φλ).Then the corresponding set of points (?)nλ={(?)ι,j: ,0≤j≤2n+1-λ,1≤ι≤λ} is a properly posed set of nodes along the set of parallel circumferences (?)(Φλ).So we extend the interpolation problem along the set of latitudes L(Φλ) to the case of the set of parallel circumferences (?)(Φλ).The following theorem describes this case.Theorem 4 Let n be a nonnegative integer and supposeλ≤n+1 is a positive integer. Then the set of points (?)nλobtained by rotation transformation is a properly posed set of nodes for Lagrange interpolation along the set of parallel circumferences (?)(Φλ) in the spaceÎ n3.Accordingly, Theorem 3 will be extended to the following one.Theorem 5 Let Vn-{Qi}i=1(n+1)2 be a properly posed set of nodes of degree n of the spaceÎ n(S2) and supposeλparallel planes intersect the sphere at a set of parallel circumferences (?)(Φλ) which does not pass through any point in Vn.Let (?)n+λ= {Qi}i=1λ(2n+2+λ) be a properly posed set of nodes of degree n+λalong (?)(Φλ).Then Vn∪(?)n+λ must be a properly posed set of nodes of degree n+λof the spaceÎ n+λ(S2).It is easy to see from above theorems that we get a class of properly posed set of nodes for Lagrange interpolation on the unit sphere by superposition interpolation process. The results obtained in this paper include not only the schemes in [56]-[58] and the schemes in [55] but also the schemes which get by superposition of the properly posed sets of nodesalong the set of parallel circumferences. Clearly, it is an extension and a development of predecessor's work in this field.Part 3. Hermite interpolation on the sphere.Suppose C1,C2,…,Cσdenoteσmutually distinct circumferences on the unit sphere and li is a unit vector which is along a diameter of the sphere and perpendicular to the circumference Ci.We denote the intersections of extension line of li and the sphere by Si and Ni and prescribe them the south pole and the north pole determined by Ci,respectively.Suppose C1,…,Cσare corresponding toσlocal coordinates which have origin as their centres of sphere and have li as their z-axis. And the spherical coordinates of Ci is ((?)i,(?)i).Choose a set of points on Ci and denote it by Vin={Qi,j,1≤j≤mi} and write Vn=∪i=1σVin$.Further more, letδi,j denote the multiplicity of each point Qi,j and assume (?)(?)δi,j=(n+1)2.We consider the following set of interpolating functionals:whereα1,α2,…,αk∈R.We call this kind of set of interpolating functionals Ln a H-class set of interpolation functionals of degree n on the sphere.Now we give the description of this kind of Hermite interpolation problem on the sphere and we call it H-class Hermite interpolation problem on the sphere.Problem 4: For any given real numbers {fi,j,k},we want to find pn(X)∈Πn(S2) such that Definition 5 If there always exists a polynomial pn(x,y,z)∈Πn(S2),such that (7) is satisfied for any given real numbers {fi,j,k},then we call Problem 4 well posed and call the set of interpolating functionals in (6) a H-class properly posed set of interpolating functionals of degree n on S2 and Vn is called the corresponding set of nodes.In order to resolve the H-class Hermite interpolation problem on the sphere, we study Hermite interpolation along a set of coaxial circumferences.Let -1i<1,1≤i≤κbeκmutually distinct real numbers andι= (cosα,cosβ,cosγ) be a given unit vector in R3.SupposeÏ€i:x cosα+y cosβ+ z cosγ= pi,(1≤i≤κ),are mutually distinct parallel planes which intersect with the sphere at circumference Ci and denote the set of coaxial circumferences which haveιas axis by C={Ci,1i denote the multiplicity ofthe circumference Ci whereδi is a positive integer.We callλ=(?)δi the totalmultiplicity of the set of coaxial circumferences and at the same time call C a set ofcoaxial circumferences includingκcoaxial circumferences whose total multiplicity isλ.Suppose the axis of a set of coaxial circumferences C whose total multiplicity isλisι.Choose a set of points Ain={Qi,j,1≤j≤mj} on each circumference Ciand write Anλ=∪i=1κAin which must satisfy (?)(?)δi,j=dim(C)=λ(2n+2-λ). We consider the following set of interpolating functionals:whereβ1,β2,…,βk∈R.Problem 5: For any given real numbers {fi,j,k},we want to find pn(X)∈Πn3such thatDefinition 6 If there always exists a polynomial pn(X)∈Πn3 such that (9) is satisfied for any given real numbers {fi,j,k,then we call Problem 5 well posed and call the set of interpolating functionals in (8) a properly posed set of interpolating functionals of degree n along the set of coaxial circumferences and An## is called the corresponding set of nodes.We have the following results for Hermite interpolation problem along the set of coaxial circumferences.Case 1.Letλ=2h+1.Supposeιis the axis of the set of coaxial circumferences C={Ci,1≤i≤κ}. Divide the equator corresponding to the axisιinto 2n+2-λequal shares and denote the great circle by (?)j(1≤j≤2n+2-λ) which pass through the j-th equidistant point on the equator, the south pole and the north pole. Then we write the set of equidistant points by Ain={Qi,j,1i,j is the intersection of Ci and (?)j and writeAnd we letδi,j=δi,mi=2n+2-λ,1≤i≤κ.We call it Class I Hermite interpolation of degree n along the set of coaxial circumferences C.Case 2.Letλ=2h.Letιbe the axis of set of coaxial circumferences C= {Ci,1≤i≤κ}.Especially, we need to impose an additional symmetry, that is, each Ci consisting of two coaxal circumferences,Ci={Ci,1,Ci,2},where Ci,1,Ci,2are symmetrical about the equator determined byι.Then we call the sum of multiplicities of Ci,1,Ci,2,1≤i≤κthe total multiplicity of C.Divide the equator determined byιinto 2(2n+2-λ) equal shares and write the great circle (?)j which pass through the j-th equidistant points on the equator, the south pole and the north pole where 1≤j≤2(2n+2-λ).We write the set of equidistant points (?)in={(?)i,j,1≤j≤2(2n+2-λ)} in which the point (?)i,j is the intersection of Ci and Cj.Write Cnλ=∪i=1κCin.Then writeAnd we letδi,j=δi,(?)2δi=λand (?)(?)δi,j=λ(2n+2-λ),mi=2n+2-λ,1≤i≤κ.We call it Classâ…¡Hermite interpolation of degree n along the set of coaxial circumferences C.Case 3. Suppose the total multiplicity of a set of coaxial circumferences C= {Ci,1≤i≤κ} isλ.For a given nonnegative integer n, consider all possible cases ofκ≤λ≤n+1 onκandλ.For each case ofκandλ,let n1 be a nonnegative integer and n2,…,nκbeκ-1 positive integers and 0≤n12<…κ=n and defineδi=ni-ni-1,where 1≤i≤κ,n0=n-λ.At the same time choose arbitrarily a set of mutually distinct pointsand writeand computeδi,j as follows:such that (?)(?)δi,j=λ(2n+2-λ).Now, in (8) we choose mi=2ni+1,1≤i≤κ.We call it Classâ…¢Hermite interpolation of degree n along the set of coaxialcircumferences C.We proved the following theorem.Theorem 6 Classâ… ,Classâ…¡and Classâ…¢Hermite interpolations of degree n along the set of coaxial circumferences C are all well posed.For H-class Hermite interpolation of degree n on the unit sphere, we give a useful superposition interpolation process, that is, the Circumferences Set-Superposition Process. Theorem 7 Let Ln be a properly posed set of interpolating functionals of degree n on the sphere and let Vn be the corresponding set of nodes. Suppose a set of coaxial circumferences C whose total multiplicity is A does not pass through any points in Vn. Let (?)n+λ,λ be a Classâ… ,or Classâ…¡,or Classâ…¢properly posed set of interpolating functionals of degree n+λalong C and let An+λλbe the corresponding set of nodes. Then Ln∪(?)n+λ,λ must be a properly posed set of interpolating functionals of degree n+λon the sphere and Vn∪An+λλbe the corresponding set of nodes.In this way, we can get a series of properly posed set of interpolating functionals of higher degree from those of lower degree on the sphere by repeatedly using the Circumferences Set-Superposition Process.Part 4. Some applications for interpolation in finite point method.The finite point method is a meshless method which gets its approximate function by using moving least square and solves equations by using matching point menthod, that is to say, to solve partial differential equation by differential method on the heterocentric set of points. The finite point method have some advantages such as directly adding the conditions on the boundary and not need to compute the integral in the interior of region.We give some notations in this section.i:the label of the point (xi,yi) or (xi,yi,zi).When we consider the especial point we often denotes it by "0";pi:the discrete value of unknown function pi-p(xi,yi) or pi=p(xi,xj,zi);Διi:the distance from point 0 to point i,Διi=(?) orΔιi=(?)Δpi:the differential of unknown function p,Δpi=pi-p0;δpi:the difference quotient of unkown function p,δpi=Δpi/Διi;(?):the vector from the point A; to the point i;li:the vector from the point k to the point i;(i,j,k):sin(?);(?):cos(?);(i,j):(i,j,0),sin(?); (?):(?):cos(?); :(?)(?),the algebraic area of the triangle consistingof the points i, j, k; (?):(?)(?); :(?)(?),the algebraic volume of thetetrahedron consisting of the points i,j, k,m;1. The case of two dimensionWe discuss the finit point method by using the directional differential quotient and directional difference quotient. We can get the interpolating polynomial using our result in previous sections. Then we can get the directional differential quotient and set up the relational expression between the directional differential quotient such that the results have a compact format. In this way we obtain the formula of two point in the numerical direction which is the foundation of constructing some kinds of computational schemes for partial differential equation on the heterocentric points.As shown in Figure 1, the neighbours of point 0:(x0,y0) are poin 1 and 2 and their dirctions l1,l2 are not parallel. Let another direction 1 and we want to approximatively compute (?) by using the values on points 0,1,2, that is, p0,p1,p2. This is a basic problem for directional differential quotient. Choosing a point 3 in the direction l and write the distance of (?) byΔι.Just as we know the choosing of point 3 is arbitrary because it is auxiliary. The order of magnitude ofΔιmust be equal to those ofΔι1 andΔι2 for the computational stability. Theorem 8 The formula for directional differential quotient of first order with two points in the direction ofιis as follows:We disperse the gradient operator and divergence operator by using the differential coefficient in the numerical direction because these operators are often used in practice. We disperse these operators using two point formula.Bacause some reasons we can not give the second order approximative formula for directional differential quotient. But we have finished some fundamental work such as to prove the well-posedness for interpolation on some points.2. The case of three dimensionAs shown in Figure 2, the neighbours of point 0:(x0,y0,z0) are poin 1, 2 and 3 and their dirctions l1,l2,l3 are not parallel. Let another direction l and we want to approximatively compute (?) by using the values on points 0,1,2,3, that is, p0,p1,p2,p3.This is a basic problem for directional differential quotient. Choosing a point 4 in the direction 1 and write the distance of (?) byΔι.Just as we know the choosing of point 4 is arbitrary because it is auxiliary. The order of magnitude ofΔιmust be equal to those ofΔιi,i=1,2,3 for the computational stability. Theorem 9 The formula for directional differential quotient of first order with three points in the direction ofιis as follows:By the same way we disperse some basic differential operators using the derived formula for differential coefficient.
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