UNIVERSITY OF WISCONSIN-MADISON COMPUTER SCIENCES DEPARTMENT. Computational aspects of polynomial interpolation in several variables

Transcription

1 UNIVERSITY OF WISCONSIN-MADISON COMPUTER SCIENCES DEPARTMENT Cmputatinal aspects f plynmial interplatin in several variables Carl de Br 1,2 & Ams Rn 1 March 1990 ABSTRACT The pair Θ,P f a pintset Θ IR d and a plynmial space P n IR d is crrect if the restrictin map P IR Θ : p p Θ is invertible, i.e., if there is, fr any f defined n Θ, a unique p P which matches f n Θ. We discuss here a particular assignment Θ Π Θ, intrduced in [3], fr which Θ, Π Θ is always crrect, and prvide an algrithm fr the cntructin f a basis fr Π Θ which is related t Gauss eliminatin applied t the Vandermnde matrix (ϑ α ) ϑ Θ,α ZZ d + fr Θ. We als discuss sme attractive prperties f the abve assignment and algrithmic details f the algrithm, and present sme bivariate examples. AMS (MOS) Subject Classificatins: primary 41A05, 41A10, 41A63, 65D05, 65D15 secndary 15A12 Key Wrds: expnentials, plynmials, multivariate, interplatin, multivariate Vandermnde, Gauss eliminatin Authrs affiliatin and address: Cmputer Sciences Department University f Wiscnsin-Madisn 1210 West Daytn St. Madisn WI supprted by the United States Army under Cntract N. DAAL03-87-K supprted by the Natinal Science Fundatin under Grant N. DMS

2 Cmputatinal aspects f plynmial interplatin in several variables Carl de Br & Ams Rn We say that the pair Θ,P f a (finite) pintset Θ IR d and a (plynmial) space P f functins n IR d is crrect if the restrictin map P IR Θ : p p Θ is invertible, i.e., if there is, fr any f defined (at least) n Θ, exactly ne p P which matches f n Θ, i.e., satisfies p(ϑ) =f(ϑ) fr all ϑ Θ. Plynmial interplatin in ne variable is s basic a Numerical Analysis tl that many textbks n Numerical Analysis begin with this tpic and nne fails t prvide a detailed accunt f it. The tpic is assciated with the illustrius names f Newtn, Cauchy, Lagrange, Hermite, and is essential fr varius basic tasks, such as the cnstructin f rules fr quadrature and differentiatin, r the cnstructin f difference apprximatins fr rdinary differential equatins. T be sure, plynmial interplatin is nt a general purpse tl fr apprximatin, fr the simple reasn that plynmials are nly gd fr lcal apprximatin (thugh fr sme particularly well-behaved functin, this might mean apprximatin n the entire line). Even in lcal apprximatin, badly handled plynmial interplatin, such as interplatin at equally spaced pints, is nt t be recmmended in general. But well-handled plynmial interplatin, such as interplatin at the Chebyshev pints, is ne f the mst efficient ways available fr lcal apprximatin. It has been therefre all the mre annying that there has nt been available a crrespndingly simple and effective thery f multivariable plynmial interplatin. The reasn is easy t spt: Whereas there is a unique interplant frm the space Π k f plynmials f degree k fr any data given n any k+1-pintset in IR, there is n crrespnding universal multivariable space f plynmials. In ther wrds, a crrect plynmial space P fr interplatin t an arbitrary f at the given set Θ IR d cannt in general be determined frm the cardinality #Θ f the pintset Θ alne. Rather, the actual lcatin and cnfiguratin f Θ must be taken int accunt. Further, the standard chice f P =Π k requires that #Θ equal ( ) k + d dim Π k (IR d ) =. d Finally, even if Θ satisfies this rather restrictive requirement, there is n guarantee that the pair Θ, Π k is crrect. In [3], we give a particular assignment Θ Π Θ fr which Θ, Π Θ is always crrect, and give an algrithm fr the cnstructin f a basis fr Π Θ frm Θ. We als prve there sme f its nice prperties. In the present paper, we list these and ther prperties f ur assignment Θ Π Θ and, eventually, verify the additinal nes. We als prvide sme enticing (s we hpe) examples. But the main pint f the present paper is a detailed discussin f the algrithmic aspects f ur particular 1

3 chice: Hw is Π Θ t be cnstructed and, nce in hand, hw is the interplant frm it t be fund? We did prvide in [3] an algrithm fr the cnstructin f Π Θ, but fund t ur surprise (cf. [2]) that Π Θ can als be cnstructed by Gauss eliminatin applied t the Vandermnde matrix ( ϑ α) fr Θ, but with a twist. This allws us t view ur particular assignment Π Θ in retrspect as arising frm a stabilizatin and symmetrizatin f a simpleminded apprach fr finding a crrect plynmial space f minimal degree fr interplatin at Θ. The paper is rganized as fllws: In Sectin 1, we recall necessary details frm [3] cncerning the definitin f ur plynmial interplant, give a very simple verificatin f ur frmula fr the interplant, and give an extensive list f its prperties. In Sectin 2, we use Gauss eliminatin t extract frm the Vandermnde matrix (ϑ α ) (fr the given ϑ Θ) a mnmial-spanned plynmial space f lwest pssible degree which is crrect fr interplatin at Θ, and prve that the same calculatin als prvides a basis fr Π Θ, albeit nt a very cnvenient ne. In Sectin 3, we shw that Gauss eliminatin applied t the Vandermnde matrix, but carried ut degree by degree rather than mnmial by mnmial, leads t a cnvenient basis fr Π Θ and prvides a suitable rdering f the pints f Θ, and cntrast this with the algrithm prpsed in [3] which crrespnds t Gauss eliminatin by clumns, with clumn pivting withut interchanges, and fails t prvide an rdering f the pints in Θ. Since, in ur multivariable setting, each degree (ther than degree 0) invlves several mnmials, we have t replace the standard gal f Gauss eliminatin, viz. the generatin f zers belw the pivt element, by the mre suitable gal f making the entries belw the pivt element rthgnal t the pivt element, with respect t a certain weighted scalar prduct. We believe that such a generalizatin f Gauss eliminatin may be advantageus in ther situatins where mre than partial pivting is needed but ttal pivting is perhaps t radical a measure. In Sectin 4, we intrduce a mdified pwer frm fr multivariate plynmials as well as a nested multiplicatin algrithm fr its efficient evaluatin. We believe bth the frm and the algrithm t be new (with the algrithm clsely related t de Casteljau s algrithm fr the evaluatin f the Bernstein-Bézier frm). In Sectin 5, we give a detailed descriptin (in a MATLAB-like prgram) f the calculatin f the mdified pwer cefficients f ur interplant frm the given data (ϑ, f(ϑ)), ϑ Θ. We illustrate the interplatin prcedure with three examples in Sectin 6: The first explres the first nntrivial case, that f a fur-pint set Θ cplanar but nt cllinear, the secnd illustrates the clse cnnectin f Π Θ t plynmials which vanish n Θ, and the third shws that the algrithm wrks sufficiently well t prvide the plynmial interplant t a smth bivariate functin at 40 randmly chsen pints. The secnd example als shws the surprising fact that ur interplant t data at the six vertices f a regular hexagn takes a cnvex cmbinatin f the given functin values as its value at every pint in a hexagn-shaped regin, and makes the pint that, fr any Θ n sme circle in the plane, ur plynmial space Π Θ cnsists f harmnic plynmials. 2

4 In Sectin 7, we prvide discussin and prfs f the varius prperties listed in Sectin 1, and clse with a shrt sectin n a generalizatin f ur prcess, frm pint evaluatins t arbitrary linear functinals n Π. Fr alternative appraches t multivariable plynmial interplatin in the literature, see, e.g., their discussin in [2]. 1. The interplant and sme f its prperties The leading term p f a plynmial p is, by definitin, the hmgeneus plynmial fr which deg(p p ) < deg p. The cnstructin prpsed in [3] makes use f an analgus cncept fr pwer series, namely the initial term f f a functin f analytic at the rigin. This is the hmgeneus plynmial f fr which f f vanishes t highest pssible rder at the rigin. In ther wrds, f (we call it f least fr shrt) is the first nntrivial term in the pwer series expansin f = f (0) + f (1) + f (2) + fr f in which f (j) is the sum f all the (hmgeneus) terms f degree j. Fr example, with ϑ x := d j=1 ϑ(j)x(j) the rdinary scalar prduct f the tw d-vectrs ϑ and x, theexpnential e ϑ with frequency ϑ has the pwer series expansin Therefre, the latter in case ϑ ϑ. We als use the abbreviatin e ϑ (x) :=e ϑ x =1+ϑ x +(ϑ x) 2 / (e ϑ ) (x) =1, (e ϑ e ϑ ) (x) =(ϑ ϑ ) x, H := span{f : f H} fr any linear space H f functins analytic at the rigin, and recall frm [3] the fact that (1.1) dim H =dimh. In these terms, ur assignment fr Π Θ is (1.2) Π Θ := (exp Θ ), with exp Θ := span{e ϑ : ϑ Θ}. Thus, if Θ cnsists f a single pint, then Π Θ =Π 0, while if Θ cnsists f the tw pints ϑ, ϑ, then Π Θ is spanned by the tw plynmials 1, (ϑ ϑ ), i.e., Π Θ cnsists f the (tw-dimensinal) space f all plynmials which are (at mst) linear in the directin ϑ ϑ and cnstant in any directin rthgnal t ϑ ϑ. 3

5 The cnstructin f ur interplant als makes use f the pairing (1.3) g, f := α D α g(0) D α f(0)/α! defined, e.g., fr an arbitrary functin g analytic at the rigin and an arbitrary plynmial f. The weights in (1.3) are chsen s that pint-evaluatin at ϑ is represented with respect t this pairing by the expnential e ϑ, i.e., (1.4) e ϑ,f = f(ϑ), ϑ IR s,f Π, as ne readily verifies by substituting ϑ α =(D α e ϑ )(0) fr (D α g)(0) in (1.3). This justifies the fllwing extensin f the pairing t arbitrary g exp Θ and f C(IR d )by w(ϑ)e ϑ,f := w(ϑ)f(ϑ), f C. ϑ Θ ϑ Θ This extensin is well defined since any cllectin f expnentials with distinct frequencies is linearly independent (see (2.4)Fact belw). Cnsequently, dim exp Θ = #Θ, and (1.5) w(i)g i = e ϑ = i i w(i) g i,f = f(ϑ) frf C. The cnstructin prpsed in [3] prvides the plynmial interplant I Θ f in the frm (1.6) I Θ f := n j=1 g j g j,f g j,g j, with g 1,g 2,...,g n a(ny) basis fr exp Θ (hence, in particular, n = #Θ) fr which (1.7) g i,g j =0 i j. Since each g j is a (hmgeneus) plynmial, it is clear that I Θ f is a plynmial. But it may be less bvius why I Θ f = f n Θ. Here is a simple argument. Frm (1.7), it fllws that I Θ f is well-defined and that (1.8) g i,i Θ f = g i,f, all i. Since g 1,g 2,...,g n is a basis fr exp Θ, this implies (with (1.4) and (1.5)) that This als implies that the space I Θ f(ϑ) = e ϑ,i Θ f = e ϑ,f = f(ϑ), ϑ Θ. span{g 1,...,g n } 4

6 is a crrect plynmial space fr interplatin at Θ. This space is cntained in Π Θ. But since dim Π Θ =dimexp Θ =#Θ=n (the first equality by (1.1)), we must have Π Θ = span{g 1,...,g n }. In rder t prvide encuragement, we nw list sme nice prperties f this particular map Θ Π Θ, but pstpne their verificatin until after the discussin f the algrithm fr the cnstructin f the interplant. (1) well-defined, i.e., fr any finite Θ, Π Θ is a well-defined plynmial space and Θ, Π Θ is crrect. (2) cntinuity (if pssible), i.e., small changes in Θ shuldn t change Π Θ by much. There are limits t this. Fr example, if Θ IR 2 cnsists f three pints, then ne wuld usually chse Π Θ =Π 1 (as ur scheme des). But, as ne f these pints appraches sme pint between the tw ther pints, this chice has t change in the limit, hence it cannt change cntinuusly. As it turns ut, ur scheme is cntinuus at every Θ fr which Π k Π Θ Π k+1 fr sme k. (3) calescence = sculatin (if pssible), i.e., as pints calesce, Lagrange interplatin appraches Hermite interplatin. This will, f curse, depend n just hw the calescence takes place. If, e.g., a pint spirals in n anther, then we cannt hpe fr sculatin. But if, e.g., ne pint appraches anther alng a straight line, then we are entitled t btain, in the limit, a match at that pint als f the directinal derivative in the directin f that line. (4) translatin-invariance, i.e., (p Π Θ,a IR d ) p(a + ) Π Θ. This implies that Π Θ is D-invariant, i.e., is clsed under differentiatin. (5) scale-invariance, i.e., (p Π Θ,α IR) p(α ) Π Θ. This is equivalent t the fact that Π Θ is spanned by hmgeneus plynmials. Nte that (4) and (5) tgether are quite restrictive in the sense that the nly finite-dimensinal spaces f smth functins satisfying (4) and (5) are plynmial spaces. (6) crdinate-system independence, i.e., an affine change f variables ϑ Aϑ+c (fr sme invertible matrix A) affects Π Θ in a reasnable way. Precisely, {invertible A IR d d,c IR d } Π AΘ+c =Π Θ A T. This implies that Π Θ inherits any symmetries (such as invariance under sme rtatins and/r reflectins) that Θ might have. This als means that Π Θ is independent f the chice f rigin. In cnjunctin with (5), it als implies that Π Θ is independent f scaling f Θ. Hence, altgether Π rθ+c =Π Θ r 0,c IR d. Finally, each p Π Θ is cnstant alng any lines rthgnal t the affine hull f Θ, i.e., Π Θ Π(affine(Θ)), with affine(θ) := { ϑ Θ ϑw(ϑ) : ϑ w(ϑ) =1}. 5

7 (7) minimal degree, i.e., the elements f Π Θ have as small a degree as is pssible. Here is the precise descriptin: Fr any plynmial space P fr which Θ,P is crrect, and fr all j, dimp Π j dim Π Θ Π j. This implies, e.g., that if Θ, Π k is crrect, then Π Θ =Π k. In ther wrds, in the mst heavily studied case, viz. f Θ fr which Π k is an acceptable chice, ur assignment wuld als be Π k. (8) mntnicity, i.e., Θ Θ = Π Θ Π Θ. This makes it pssible t develp a Newtn frm fr the interplant. Als, in cnjunctin with (7) and (9), this ties ur scheme clsely t standard chices. (9) Cartesian prduct = tensr prduct, i.e., Π Θ Θ = Π Θ Π Θ.Inthisway, ur assignment in the case f a rectangular grid cincides with the assignment standard fr that case. In fact, in cnjunctin with (8), we can cnclude that we btain the standard assignment even in the case that Θ is a lwer set f a rectangular grid f pints (see Sectin 7). (10) assciated differential peratrs. This unusual prperty links plynmials p which vanish n Θ t hmgeneus cnstant cefficient differential peratrs q(d) which vanish n Π Θ. The precise statement is that such q(d) vanishes n Π Θ if and nly if the hmgeneus plynmial q is the leading term p f sme plynmial p which vanishes n Θ. We expect this prperty t play a majr rle in frmulae fr the interplatin errr. (11) cnstructible, i.e., a basis fr Π Θ can be cnstructed in finitely many arithmetic steps. This list prvides enugh details t make it pssible t identify Π Θ in certain simple situatins directly, withut the aid f the defining frmula (1.2). Fr example, if #Θ = 1, then necessarily Π Θ =Π 0 (by (7)). If #Θ = 2, then, by (6) and (7), necessarily Π Θ = Π 1 (affine(θ)). If #Θ = 3, then Π Θ =Π k (affine(θ)), with k := 3 dim affine(θ). The case #Θ = 4 is the first ne that is nt clear-cut. In this case, we have again Π Θ =Π k (affine(θ)), k := 4 dim affine(θ), but nly fr k =1, 3. When affine(θ) is a plane, we may use (6) t nrmalize t the situatin that Θ IR 2 and Θ = {0, (1, 0), (0, 1),θ}, with θ, ffhand, arbitrary. Since Π 1 is the chice fr the set {0, (1, 0), (0, 1)}, this means that Π Θ =Π 1 + span{q} fr sme hmgeneus quadratic plynmial q. While (2) and (6) impse further restrictins, it seems pssible t cnstruct a suitable map θ q in many ways s that the resulting Θ Π Θ satisfies all the abve cnditins, except cnditins (8) and (10) perhaps. (See Sectin 6 fr ur chice fr q = q θ.) At present, we d nt knw whether there is nly ne map Θ Π Θ satisfying all cnditins (1)-(9). But, additin f cnditin (10) uniquely determines the map. Of curse, we didn t make up the abve list and then set ut t find the map Θ Π Θ. Rather, we came acrss the fact that the pair Θ, (exp Θ ) is always crrect, and this started us ff studying the assignment Π Θ := (exp Θ ). 6

8 2. The chice f P prvided by eliminatin In this sectin, we prvide further insight int ur particular assignment Π Θ =(exp Θ ) by cmparing it with a mre straightfrward assignment which is prvided by Gauss eliminatin applied t the Vandermnde matrix fr Θ. This als shuld help in the understanding f the algrithm fr the cnstructin f I Θ described in the next sectin. In the absence f bases fr the space Π := Π(IR d ) f all plynmials in d variables mre suitable fr calculatins with multivariable plynmials, we deal here with the pwer frm, i.e., we express plynmials as linear cmbinatins f the pwers () α :IR d IR : x x α := x α(1) 1 x α(d) d. The plynmial p =: α ()α c(α) nir d matches the functin f at the pintset Θ if and nly if its cefficient sequence c := ( c(α) ) slves the linear system α ZZ d + (2.1) V?=f Θ, with (2.2) V := ( ϑ α) ϑ Θ,α ZZ d + the Vandermnde matrix fr Θ. Thus a search fr plynmial interplants t f at Θ is a search fr slutins c :ZZ d + IR f (2.1) f finite supprt (i.e., with all but finitely many entries equal t zer). Actual calculatins wuld frce us t rder the pints in Θ and the indices α ZZ d +. It is mre cnvenient, thugh, t let the ϑ Θ and the α ZZ d + index themselves fr the time being. Thus V is a linear map taking functins n ZZ d + t functins n Θ. Its clumns crrespnd t α ZZ d +, its rws t ϑ Θ. (2.3)Prpsitin. The Vandermnde matrix V (see (2.2)) is f full rank. Prf: One way t see this is t bserve that (a(ϑ)) ϑ Θ V = 0 implies that ϑ Θ a(ϑ)e ϑ = 0 (since ϑ α =(D α e ϑ )(0)), and thus t rely n the fllwing (2.4)Fact. Any cllectin f expnentials with distinct frequencies is linearly independent. Prf: The prf is by inductin since the linear independence is bvius when #Θ=1. If#Θ> 1 and s := ϑ Θ a(ϑ)e ϑ = 0 with all a(ϑ) 0, then als (D y c)s = ϑ Θ ((y ϑ) c)a(ϑ)e ϑ = 0 fr any particular y. Since the ϑ are distinct, we can chse y and c s that y θ = c fr a particular θ Θ while y ϑ c fr at least ne ϑ Θ. Thus ϑ θ ((y ϑ) c)a(ϑ)e ϑ = 0 is a sum f the same nature but with ne fewer summand, hence with all its cefficients zer by inductin hypthesis, hence at least ne f the a(ϑ) must be zer, cntrary t ur assumptin. 7

9 Eliminatin is the standard tl prvided by Linear Algebra fr the determinatin f the slutin set f any linear (algebraic) system. Eliminatin classifies the unknwns int bund and free. Assuming the cefficient matrix t be f full rank (which ur matrix V is by (2.3)Prpsitin), this means that each rw is designated a pivt rw fr sme unknwn, which thereby is bund, i.e., cmputable nce all later unknwns are determined. Any unknwn nt bund is free, i.e., freely chsable. Standard eliminatin prceeds in rder, frm left t right and frm tp t bttm, if pssible. In Gauss eliminatin with partial pivting, ne insists n prceeding frm left t right, but is willing t rearrange the rws, if necessary. Thus, Gauss eliminatin with partial pivting applied t (2.1) (written accrding t sme rdering f the ϑ Θ and the α ZZ d +) prduces a factrizatin LW = V, with L unit lwer triangular and W in rw echeln frm. This means that there is a sequence β 1,β 2,...,β n which is strictly increasing, in the same ttal rdering f ZZ d + that was used t rder the clumns f V, and s that, fr sme rdering {ϑ 1,ϑ 2,...,ϑ n } f Θ and fr all j, the entry W (ϑ j,β j ) is the first nnzer entry in the rw W (ϑ j, :) f W. (2.5)Prpsitin. Let LW = V be the factrizatin f V prvided by Gauss eliminatin with partial pivting. Specifically, let β 1,β 2,...,β n be the sequence, strictly increasing in the same ttal rdering f ZZ d + that was used t rder the clumns f V, fr which, fr sme rdering {ϑ 1,ϑ 2,...,ϑ n } f Θ and fr all j, the entry W (ϑ j,β j ) is the first nnzer entry in the rw W (ϑ j, :) f W. Then P := span(() β j ) n j=1 is crrect fr interplatin at Θ. Mrever, if the clumns f V are rdered by degree, then P is a plynmial space f smallest pssible degree which is crrect fr Θ. Prf: By assumptin, the square matrix ( ) n U := W (ϑ i,β j ) is upper triangular and invertible, and s prvides the particular interplant i ()β i a(i), whse cefficient vectr i,j=1 (2.6) a := (LU) 1 (f(ϑ 1 ),...,f(ϑ n )) is btainable frm the riginal data f Θ by permutatin fllwed by frward- and backsubstitutin. Nw recall that Gauss eliminatin determines the next pivt clumn as the clsest pssible clumn t the right f the present pivt clumn. This means that each β j is chsen as the smallest pssible index greater than β j 1, in whatever rder we chse t write dwn the clumns f V. Cnsequently, the plynmial space ( ) n P := span () β i selected by this prcess is spanned by mnmials f smallest pssible expnent (in the rdering f ZZ d + used). In particular, assume that we rdered the α by degree, i.e., s that i=1 α<β = α β, 8

10 with α := α(1) α(d) the custmary abbreviatin fr the length f the index vectr α. Then P is f smallest degree (since eliminatin applied t a matrix f full rank determines the shrtest initial segment f full rank f that matrix). Nw nte that the plynmial space P cnstructed in the prpsitin may well change drastically in respnse t a simple change f variables. Fr example, if Θ = {(0, 0), (1, 0)} IR 2, and we use the standard rdering (0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2),... fr ZZ 2 +, then, fr any rtate AΘ f Θ, eliminatin wuld prvide the space span{() 0, () 1,0 }, except fr rtatin by 90 degrees, in which case span{() 0, () 0,1 } wuld be selected. This simple example als illustrates that the sequence β 1,β 2,...,β n need nt turn ut t cnsist f cnsecutive terms, even if we rdered the α by α. (Facts like this have prevented the develpment f a simple thery f multivariate plynmial interplatin.) Rather, eliminatin has t face the numerical difficulty f deciding when all the pivts available fr the current step in the current clumn are practically zer, in which case the pivt search is extended t the entries in the next clumn (and in any rw nt yet used as pivt rw). But this can als be viewed psitively. Just as partial rw pivting has as its gal the smallness f the factrs L and U, s the additinal freedm f clumn pivting allwed here prvides further means fr keeping the factrs L and U small. The smaller these factrs, the better is the cnditin f the crrespnding basis ( ) () β j fr the plynmial space P selected, when cnsidered as a space f functins n Θ. (2.7)Therem. Assume the clumns f the Vandermnde matrix V = ( ϑ α) rdered by degree and let LW = V be the factrizatin f V prvided by Gauss eliminatin with partial pivting. Specifically, let β 1,β 2,...,β n be the strictly increasing sequence fr which, fr sme rdering {ϑ 1,ϑ 2,...,ϑ n } f Θ and fr all j, the entry W (ϑ j,β j ) is the first nnzer entry in the rw W (ϑ j, :) f W. Then h i := α = β i W (ϑ i,α)() α /α!, i =1,...,n is basis fr Π Θ. Prf: Since V (ϑ, α) =ϑ α =(D α e ϑ )(0), and W = L 1 V, it fllws that each g i := α W (ϑ i,α)() α /α!isinexp Θ, hence h i := α = β i W (ϑ i,α)() α /α! =g i is in Π Θ = (exp Θ ). Since W is in rw-echeln frm (specifically, W (ϑ i,β i ) is the first nnzer entry in the rw W (ϑ i, :) and the sequence β i is strictly increasing), we knw that h 1,h 2,...,h n is linearly independent, hence a basis fr Π Θ, since dim Π Θ = n by (1.1). Thus, the assignment Θ Π Θ prpsed in [3] turns ut t differ frm the naive assignment made in (2.5)Prpsitin in nly ne (imprtant) detail: Instead f the space spanned by the particular mnmials () β i singled ut by eliminatin, we take the space spanned by the least terms g i f the functins g i := α W (ϑ i,α)() α /α!. But these particular g 1,g 2,...,g n d nt in general satisfy (1.7). T btain a basis g 1,g 2,...,g n fr exp Θ satisfying (1.7), we carry ut eliminatin, nt mnmial by mnmial, but degree by degree. 9

11 3. Eliminatin by degree We prpsed in [3] a particular algrithm fr the cnstructin f the basis g 1,g 2,..., g n fr exp Θ satisfying (1.7) and needed fr (1.6). But, with the details f Gauss eliminatin recalled in the preceding sectin in mind, it seems mre efficient t cnstruct (as already prpsed in [2]) theg i by applying Gauss eliminatin with partial pivting t the matrix V := (ϑ k) btained frm the Vandermnde V = ( ϑ α) by treating all entries f a given degree as ne entry. We have written V instead f V t signify this alternate pint f view. Thus V has its rws indexed by ϑ Θ as befre, but its clumns are indexed by k =0, 1, 2,... Crrespndingly, ϑ k = ( ϑ α) α =k. Since the entries f V are vectrs, rather than just numbers, we cannt hpe t eliminate entries, we can nly hpe t make all the entries in the pivt clumn belw the pivt rw rthgnal t the pivt entry. Because f (1.3), the relevant scalar prduct is (3.1) a, b k := a(α)b(α)/α! α =k when eliminating in clumn k f V. In rder t keep the ntatin uncluttered, we will use the abbreviatin W(ϑ, k), W(ϑ, k) := W(ϑ, k), W(ϑ, k) k, with W any matrix which, like V, has vectrs indexed by {α ZZ d : α = k} as the entries in its kth clumn. It fllws that a given clumn may be pivt clumn fr several pivt rws. Still, the verall prcess f Gauss eliminatin with partial pivting applied t a matrix like V is clear: Let W be the wrking array which initially equals V. Atthejth step, we lk fr the smallest k j k j 1 fr which there is a nntrivial entry f W in clumn k j at r belw rw j. Then we find a largest such entry (relative t the size f the crrespnding entry r rw f V) and, if necessary, interchange its rw with rw j f W t bring it int the pivt psitin W(ϑ j,k j ). Then we subtract the apprpriate multiple f the pivt rw W(ϑ j, :) frm all subsequent rws in rder t make W(ϑ i,k j ) rthgnal t W(ϑ j,k j ) fr all i>j. The result is a factrizatin LW = V, with L again unit lwer triangular, but W is in rw echeln frm in the fllwing sense. There is a nndecreasing sequence k 1,k 2,...,k n and sme rdering {ϑ 1,ϑ 2,...,ϑ n } f Θ s that, fr all j, the (vectr-)entry W(ϑ j,k j ) is the first nnzer entry in the rw W(ϑ j, :) f W. In ther wrds, the matrix ( ) n W(ϑ i,k j ) i,j=1 10

12 is blck upper triangular, with nnzer diagnal entries. Nte that this matrix need nt be upper triangular, since the sequence k 1,k 2,...,k n need nt be strictly increasing. But, there has t be rthgnality f W(ϑ i,k j )tw(ϑ j,k j ) when k i = k j and i j. Explicitly, the square matrix ( ) n (3.2) U := W(ϑ i,k j ), W(ϑ j,k j ) i,j=1 is upper triangular and invertible. Cnsequently, with UG := W, the matrix ( ) n G(ϑ i,k j ), G(ϑ j,k j ) i,j=1 is diagnal and invertible. Fr, factring ut the upper triangular matrix U is equivalent t backward eliminatin, i.e., t the calculatins fr j = n, n 1,...,1, d: W (ϑ j, :) W (ϑ j, :)/U(j, j) fr i =1,...,j 1, d: W (ϑ i, :) W (ϑ i, :) U(i, j)w (ϑ j, :) end end in which the jth step enfrces rthgnality f the pivt element in rw j t the elements abve it in the pivt clumn, withut changing the rthgnalities already achieved in subsequent clumns, and withut changing anything in the preceding clumns. Thus, in terms f the weighted scalar prduct (3.3) a, b := k a, b k = α a(α)b(α)/α! fr sequences a, b :ZZ d + IR, we have (3.4) G(ϑ i, :),G kj (ϑ j, :) = δ i,j /U(j, j), i,j =1,...,n, with G k given by With this, let G k (:,α):= { G(:,α), α = k; 0, therwise. (3.5) g i := α () α /α! G(ϑ i,α). Then (3.6) (LU)(i, j)g j = e ϑj, all j, j 11

13 since LUG = V = ( D α e ϑi (0) ) i,α. Further, (3.7) g i = α =k i () α /α! G(ϑ i,α) = () α /α! G ki (ϑ i,α), α and we cnclude frm (3.4) that (3.8) g i,g j = δ i,j /U(j, j), i,j =1,...,n. Since g 1,g 2,...,g n is linearly independent by (3.8), (3.6) implies that g 1,g 2,...,g n is a basis fr exp Θ. But (3.8) als implies that g 1,...,g n s cnstructed is linearly independent, hence a basis fr Π Θ by (1.1). This prves (3.9)Therem. The functins g i defined by (3.5) prvide a basis fr exp Θ which satisfies (1.7), and the crrespnding g i frm a basis fr Π Θ. (3.10)Crllary. Let (3.11) a := diag(u)(lu) 1 (f(ϑ 1 ),...,f(ϑ n )), with L, U, and ϑ 1,ϑ 2,...,ϑ n determined during Gauss eliminatin with partial pivting applied t V as described abve. Then, with g i as given by (3.7), I Θ f = j g j a(j) is the unique interplant frm Π Θ t f n Θ. Prf: The functin q := j g j a(j) isinπ Θ by (3.9)Therem. Further, frm (3.8), g j,q = a(j)/u(j, j). Therefre, frm (3.6) and (3.11), q(ϑ i )= e ϑi,q = j LU(i, j) g j,q = j LU(i, j) r (LU) 1 (j, r)f(ϑ r )=f(ϑ i ). In effect, the multiplicatin in (3.11) by the diagnal matrix diag(u) accunts fr the divisin by g j,g j in (1.6), as the latter number is 1/U(j, j), by (3.8). It is wrth nting that the factring ut f U frm W will nt change the pivt entries W(ϑ j,k j ) since U(i, j) =0ifk i = k j and i j, except fr the nrmalizing divisin. In ther wrds, = W ki /U(i, i), G ki (with W k defined entirely analgusly t G k ), shwing that the factring ut f U frm W need nt be carried ut, unless ne is interested in the g i rather than the g i. On the ther hand, frmatin f U is essential fr the calculatin f the cefficients f the interplating plynmial. 12

14 In the language intrduced in this sectin, the algrithm fr the calculatin f suitable g 1,g 2,...,g n frm f j := e ϑj, j =1,...,n prpsed in [3] amunts t Gauss eliminatin with clumn pivting applied t V, except that n clumns are actually interchanged. Rather, at the jth step, ne lks fr the left-mst nnzer entry in the jth rw f the wrking array W, saytheentryw(ϑ j,k j ), then uses the jth rw t make all entries W(ϑ i,k j )fri j rthgnal t W(ϑ j,k j ). This will nt spil rthgnality f W(ϑ i,k i ) t W(ϑ r,k i )frr i and i<jachieved earlier, since either k i <k j, hence W(ϑ j,k i )=0, r k i >k j, hence W(ϑ j,k j ) is trivially rthgnal t W(ϑ i,k j ) = 0, r k i = k j, hence W(ϑ j,k j ) is already rthgnal t W(ϑ i,k j ). Thus ne btains a factrizatin AW = V, with A invertible, and W in reduced rw echeln frm in the sense that, fr sme sequence k 1,k 2,...,k n, W(ϑ i,k j ), W(ϑ j,k j ) kj =0 i j. This implies that the functins g 1,g 2,...,g n defined by g j := α () α /α! W (ϑ j,α) satisfy (1.7), hence the crrespnding g j must be a basis fr Π Θ (by the reasning used earlier). It is nt bvius withut recurse t the results frm [3] that the tw sequences g 1,...,g n prduced by the tw algrithms span the same space. The algrithm utlined in this sectin seems preferable t the ne frm [3] nt nly because it is clser t a standard algrithm but als because it prvides a ready means fr rdering the pints f Θ fr greater stability f the calculatins. Sme f the finer cmputatinal details are taken up belw, after a shrt sectin n a particularly suitable plynmial frm. 4. Nested multiplicatin fr the mdified pwer frm We knw nly tw plynmial frms readily available fr the presentatin f plynmials in several variables, the pwer frm and the Bernstein-Bézier frm. The calculatins abve are in terms f the pwer frm p = α () α D α p(0)/α!, hence we stick with that frm here, particularly since we are nt cncerned here with the Bernstein-Bézier frm s majr strength, the smth patching f plynmial pieces (see, e.g., [1]). It is nly prudent t use the shifted pwer frm, i.e., t write p = α ( c) α D α p(c)/α!, 13

15 fr sme apprpriate center c, e.g., c = c Θ := ϑ Θ ϑ/#θ. Equivalently, we assume that Θ has been shifted at the utset by its center c Θ. It turns ut t be simpler t use the fllwing mdified pwer frm (4.1) p = ( ) α () α D α p(0)/ α!, α α with ( ) α α := α!/α! the multinmial cefficients. There are tw reasns. (i) It is easy t prgram and use the fllwing multivariable versin f nested multiplicatin (r Hrner s scheme): (4.2)Prpsitin. If { D α p(0)/ α!, α = deg p; (4.3) c(α) := D α p(0)/ α!+ d i=1 x ic(α + i i ), α = deg p 1, deg p 2,...,0, with i i the ith unit vectr, p Π(IR d ), and x IR d, then c(0) = p(x). Prf: Indeed, it fllws that c(0) = α deg p n α x α D α p(0)/ α!, with n α the number f different increasing paths t α frm the rigin thrugh pints f ZZ d +. This number is n α = ( ) α α, hence c(0) = p(x), by (4.1). In effect, it is pssible t evaluate a multivariable plynmial p frm its nrmalized Taylr cefficients (D α p)(0)/ α! withut the (explicit) cmputatin f multinmial cefficients. (ii) The infrmatin abut g j cmputed by the algrithm utlined in the preceding sectin readily prvides the numbers D α g j (0) (see (3.5)), hence the calculatin f the mdified pwer frm fr I Θ f, i.e., f the nrmalized Taylr cefficients (D α I Θ f)(0), frm the matrix G and the vectr (LU) 1 f Θ can be accmplished withut generatin and use f the multinmial cefficients. The clse similarity t de Casteljau s algrithm (see, e.g., [1]) fr the evaluatin f the Bernstein-Bézier frm is actually nt surprising, fr the fllwing reasn. The Bernstein- Bézier frm x ( ) β (ξ(x)) β c(β) β β =k describes a plynmial f degree k in terms f the d + 1 linear plynmials ξ t defined by the identity ξ t p(t) =p fr all p Π 1, t T 14

16 with T IR d in general psitin. The de Casteljau algrithm fr its evaluatin at sme x cnsists f the calculatins c(β) := t T ξ t (x)c(β + i t ), β = j, fr j = k 1,k 2,...,0, with the resulting c(0) the desired value at x. While the vectr ξ(x) prvides the barycentric crdinates f x with respect t the pintset T, n use is made in the de Casteljau algrithm f the fact that t T ξ t(x) = 1. Thus the calculatins c(α) := d x i c(α + i i ), α = j, i=1 fr j = k 1,k 2,...,0 and started frm given c(α) with α = k will prvide the number α =k ( ) α x α c(α). α The full algrithm abve merely cmbines apprpriately the steps cmmn t de Casteljau applied t the terms in (4.1) f different degrees. 5. Algrithmic details We give here a (smewhat infrmal) MATLAB-like prgram (see, e.g., [6] fr language details) fr the cnstructin f ur interplant in rder t dcument the simplicity f the actual calculatins needed. In this prgram, we use the fllwing cnventins: V and W dente the matrices V and W, respectively. In particular, W(i,k) is a vectr with ( ) k+d 1 d 1 entries, indexed by {α ZZ d : α = k}. This is decidedly nt allwable in present-day MATLAB, but cnvenient here, as it avids discussin f the (imprtant technical) questin f the best way t rder the index set {α ZZ d : α = k}. Crrespndingly, fr tw vectrs a and b (such as W(i,k), W(j,k)) indexed by {α ZZ d : α = k}, <a,b> dentes the (scaled) scalar prduct (5.1) <a,b> := α =k ( ) α a(α)b(α) α related t (3.1) (with k = k). All matrices mentined in the prgram ther than V and W are prper MATLAB matrices, i.e., have scalar entries. Further, we use a <-- b t indicate that a is t be verwritten with the cntents f b, and use an ccasinal English wrd r tw t describe an actin whse details seem clear. We brrw frm MATLAB the ntatins: (i) eye(n,n) fr the identity matrix f rder n; (ii) nes(m,n) fr the matrix f size m n with all entries equal t 1; (iii) zers(m,n) fr the matrix f size m n with all entries equal t 0; (iv) a:b fr the vectr with entries 15

17 a, a +1,...,a + m, with m the natural number fr which a + m b < a + m +1; (v) A*B fr the matrix prduct f the matrices A and B; (vi) standard lgical cnstructs like (fr j=1:n,...,end), and (if...,..., end); (vii) the cnstruct (while 1,..., if..., break, end,..., end), which is a lp exited nly thrugh the break; (viii) the cnstruct [m,i] <-- max(a) t prvide m := a(i) :=max j a(j). % INPUT: Θ={ϑ 1,...,ϑ n }, f Θ, tl k <-- 0 V(:,k) <-- nes(n,1); W(:,k) <-- V(:,k) L <-- eye(n,n); U <-- zers(n,n); K <-- zers(1,n) fr j=1:n while 1 [m,i] <-- max{<w(i,k),w(i,k)>/<v(i,k),v(i,k)>: i>j-1} if m>tl, break, end k <-- k+1 V(:,k) <-- frm V(:,k-1) and Θ W(:,k) <-- L 1 *V(:,k) end if i>j, interchange rws i and j, end K(j) <-- k fr i=1:j, U(i,j) <-- <W(i,k),W(j,k)> end fr i=j+1:n L(i,j) <-- <W(i,k),W(j,k)>/U(j,j) W(i,k) <-- W(i,k) - L(i,j)*W(j,k) end end W <-- U 1 *W % f <-- prperly permuted f Θ a <-- diag(u)*u 1 *L 1 *f kmax <-- max(k); dk <-- ( ) [0:kmax]+d 1 d 1 cefs <-- zers(kmax+1,dk(kmax)) fr j=1:n range=1:dk(k(j)) cefs(k(j),range) <-- cefs(k(j),range) + a(j)*w(j,k(j)) end % OUTPUT: cefs The utput prvides the cefficients c(α) =cefs( α,α) fr the mdified pwer frm f the interplant. The needed weights w(α) := ( ) α α fr the (scaled) scalar prduct (5.1) are integers and are cnveniently generated frm their recurrence relatin d (5.2) w(α) = w(α i i ) i=1 16

18 each time k is increased by 1, using the initial value and the side cnditins w(0) = 1 w(α) =0 α ZZ d +. The algrithm des require a sensible rdering f the index set {α : α = k} in rder t facilitate (i) use f (5.2); (ii) the efficient calculatin f the entries f V(ϑ, k) frm thse f V(ϑ, k 1), e.g., via (5.3) V (ϑ, α) =ϑ i V (ϑ, α i i ), with α s.t. α(j) =0frj<i; and fr i =1,...,d; and (iii) the use f (4.3). We have used the inverse lexicgraphic rdering f the α. The utput des depend n the chice f the tlerance tl. In exact arithmetic, we culd chse tl =0. (5.4)Prpsitin. If the abve algrithm is run in exact arithmetic with tl =0, then the while lp is never repeated. Prf: In exact arithmetic and with tl = 0, the algrithm prvides a hmgeneus basis fr Π Θ, by (3.9)Therem, and the plynmial degrees f these basis elements are the numbers k appearing in the prgram. Under certain cnditins, this number is increased by 1 in the while lp. If a secnd increase were necessary fr the current j, then it wuld fllw that there is a gap in the degrees f sme hmgeneus basis fr Π Θ, and this wuld cntradict the D-invariance f Π Θ (see (4)). In finite-precisin arithmetic, the chice f tl is mre delicate. It shuld reflect the number f digits carried during the calculatins. As tl is increased, we can expect L and U t be f smaller size, but may eventually nt btain a plynmial space clse t Π Θ. This can be f cnsiderable numerical advantage in the case that a zer tlerance wuld lead t an unacceptably large U. This is analgus t the cmmn practice f treating a cluster f (simple) zers f a functin numerically as ne zer f apprpriate multiplicity. 6. Examples (6.1)Fur pints in the plane. We start with the simplest nntrivial Θ, viz. a Θ made up f fur pints whse affine hull is a plane. As already pinted ut in Sectin 1, we may assume withut lss that Θ = {0, i 1, i 2, (u, v)}, with i j the jth unit vectr in IR 2. With this rdering f the pints, and the lexicgraphic rdering fr the α, the Vandermnde matrix becmes V = u v u 2 uv v

19 Eliminatin (withut pivting) with the scaled scalar prduct (5.1) generates the matrices L =, W = u v u 2 u uv v 2 v... Hence u(u 1) U =, w := (u 2 + v 2 ) 2 2(u 3 + v 3 )+(u 2 + v 2 ) v(v 1) w This gives u G = 2 (u 1) 2 /w u 2 (u 1)v/w u(u 1)v(v 1)/w u(u 1)v(v 1)/w uv 2 (v 1)/w 1 v 2 (v 1) 2. /w u(u 1)/w uv/w v(v 1)/w... The resulting basis fr Π Θ cnsists f g 1 =() 0, g 2 =() 1,0, g 3 =() 0,1, g 4 =() 2,0 u(u 1)/(2w)+() 1,1 uv/w +() 0,2 v(v 1)/(2w), with g i,g 4 =(1/w)δ i4. This illustrates the fact that, fr the purpse f cnstructing the interplant, there is n need t cnstruct the matrix G. Even fr this simple example, the resulting frmulae are nt particularly simple r pretty. On the ther hand, there is n suggestin here t carry ut such calculatins by hand. On the third hand, it is easy t see in this simple example what happens as pints becme cllinear. E.g., if v 0, g 4 simplifies t g 4 =() 2,0 /2(u(u 1)), i.e., Π Θ nw cntains Π 2 (IR {0}), as it shuld. But this wrks ut nly if u {0, 1}. If (u, v) 0r(u, v) (1, 0), then the furth pint (u, v) wuld be appraching the first r secnd pint in Θ and the limiting Π Θ nw will depend n just hw this apprach is made. (6.2)Hexagn pints. Because f the inherent symmetries, the frmula fr interplatin at the vertices f a regular hexagn is very pretty indeed. Assume withut lss that Θ={ϑ j := (cs(t j ), sin(t j )) : j =1,...,6}, with t j := 2πj/6, all j. 18

20 Since dim Π 2 (IR 2 ) = 6, we expect Π Θ =Π 2 (IR 2 ) fr the generic 6-pint set Θ in the plane. But the hexagn pints lie n the unit circle, i.e., the plynmial p := 1 () 2,0 () 0,2 Π 2 (IR 2 ) vanishes n Θ, hence Θ, Π 2 (IR 2 ) cannt be crrect. Further, by (10), p must be rthgnal (in the sense f the pairing (1.3)) t Π Θ. On the ther hand, any five f these six pints are generic, i.e., they are nt cllinear. Hence Π Θ =(Π 2 span(p )) + span(q), with the rthgnal cmplement (Π 2 span(p )) f span(p )inπ 2 taken in the sense f the pairing (1.3), and with q a certain hmgeneus third-degree plynmial. Here is ne way t determine this q: Cnsider the interplant I Θ f t data f(ϑ j )= ( ) j,allj. There are three lines thrugh the rigin nt cntaining any f the interplatin pints but such that reflectin acrss that line leaves Θ unchanged. By (6), such reflectin must als leave Π Θ unchanged. On the ther hand, it will map the data t their negative. This implies that such reflectin must map I Θ f t its negative, cnsequently I Θ f must vanish alng each f these three lines. Therefre, I Θ f vanishes t secnd degree at the rigin, hence is a hmgeneus cubic, hence q = I Θ f (since q is determined nly up t scalar multiples, anyway). In particular, rtatin by π/3 maps q t its negative. Anther way is t nte that Θ is unchanged under rtatin by π/3, hence such rtatin must map q t rq fr sme real r fr which r 6 = 1. The chice r = 1 wuld lead t the cnclusin that q is cnstant n Θ, therefre cnstant thrughut, by uniqueness f the interplant, and this wuld cntradict the fact that q is a third-degree plynmial. Thus, necessarily, r = 1, i.e., rtatin by π/3 maps q t its negative. This is the same cnclusin reached in the preceding paragraph. It fllws that q =() 3,0 3() 1,2 =Rez 3, with z := () 1,0 + i() 0,1 the cmplex independent variable. This is related t the fact that I Θ f is the real part f the (cmplex) Lagrange interplant t the data (f(ϑ j )) j at the six rts f unity in the cmplex plane. (Since z 3 takes the value ( ) j at the pint ϑ j (1) + iϑ j (2), z z 3 is an interplant frm Π 5 (C), therefre the interplant). As a matter f fact, Prperty (10) implies that, fr any Θ n any particular circle, Π Θ must cnsist f harmnic plynmials since then Θ is mapped t zer by sme plynmial whse leading part is () 2,0 +() 0,2. In particular, fr any six-pint set Θ n a circle, the hmgeneus cubic plynmial in Π Θ is a linear cmbinatin f Re z 3 and Im z 3. The three heavy lines in (6.3)Figure shw the zers f the (real) Lagrange plynmial l 6 fr ur interplatin at the hexagn pints. This means that l 6 is the unique plynmial in Π Θ whichis1at(1, 0) and 0 at the ther five pints. Frm the figure (and the fact that l 6 (1, 0) = 1), we cnclude that l 6 is psitive near 0. Since Π Θ is invariant under rtatin by π/3, the remaining five Lagrange plynmials l j, j =1,...,5, are btained frm l 6 by rtatin by the apprpriate multiple f π/3, hence are als psitive near the rigin. Since the Lagrange plynmials sum t the cnstant functin 1, this implies the fllwing surprising fact. 19

21 (6.3)Figure. The Lebesgue functin fr interplatin at the hexagn pints is ne n the entire central prtin (6.5)Figure. Cntur lines, fr values 1, 1.05,..., 2, f Lebesgue functin fr interplatin at hexagn pints and their center. (6.4)Prpsitin. In the hexagnal dmain utlined by the zersets f the l j,thevalue f the interplant is a cnvex cmbinatin f the given functin values. On the ther hand, as the radius f the circle shrinks t zer, the interplant I Θ f apprximates f near 0 nly t first rder since the prcess fails t reprduce all f Π 2.In 20

22 rder t remedy this, we enlarge Θ by adding the rigin t it. This is bund t destry the psitivity near the rigin f the Lagrange plynmials fr the ther pints (and makes p =1 () 2,0 () 0,2 the Lagrange plynmial fr the seventh pint, with the remaining Lagrange plynmials btained frm thse fr the hexagn pints by subtractin f p/6). But the Lebesgue functin fr the resulting interplatin prcess, i.e., the sum f the abslute values f all the Lagrange plynmials, stays remarkably clse t 1 near the rigin. (6.5)Figure shws the level lines f the Lebesgue functin, fr the values 1:.05:2. This shws that, n the unit circle, the Lebesgue functin des nt exceed Figure 4. Cntur lines fr errr in interplatin at 40 randm pints. (6.6)Randm pints In this example, we chse 40 pints in the square [0..1] 2 at randm as interplatin pints, and cnstructed an interplant at these pints t the functin x exp( x 2 1 x 2 2). (6.5)Figure shws ten level lines (crrespnding t ten equal values between the minimum and the maximum) f the errr in the resulting interplant. In particular, (6.5)Figure makes clear that the errr is clse t zer in an area arund the interplatin pints. The abslutely largest errr turned ut t be 3e 4. We fund that this example was nt at all islated. 7. Verificatin f list f prperties In this sectin, we verify the varius prperties asserted fr ur interplatin scheme in Sectin 1. The first eight prperties are either evident frm ur definitin f Π Θ r are dealt with in detail in [3]. But sme f the cnsequences mentined still require prf. 21

23 We begin with a derivatin f the affine hull prperty Π Θ Π(affine(Θ)). First we give a prf based n (6), and then fllw this with shrter prfs based n the strnger prperties (9) and (10). After a rigid mtin, we may assume that affine(θ) = IR s {0}. Since Π Θ is invariant under any linear change f variables which leaves Θ unchanged, it is in particular invariant under any scaling f the arguments s +1,...,d. This implies that Π Θ must cntain any plynmial p which des nt depend n its arguments s +1,...,d and agrees n IR s {0} with sme plynmial frm Π Θ. On the ther hand, the restrictin map p p IR s {0} must be 1-1 n Π Θ, since Θ IR s {0} and the restrictin map p p Θ is 1-1. The affine hull prperty can als be derived frm the prperty (9) cncerning tensr prducts, since, after that linear change f variables which makes affine(θ) = IR s {0}, we may think f Θ as Θ = (Θ IR s {0}) ({0} IR d s ). Finally, the affine hull prperty can als be written D y (Π Θ ) = 0 fr all y affine(θ), i.e., fr all y fr which the linear plynmial x y x is cnstant n Θ. Thus this prperty is a very special case f (10). Next, we nte that the minimal degree prperty (7) is equivalent t the prperty (7 ) degree-reducing, i.e., fr every plynmial p, deg I Θ p deg p. (7.1)Prpsitin. Let Θ,P be crrect, with P a plynmial space, and let I be the crrespnding interplatin map. Then P is f minimal degree if and nly if deg Ip deg p fr all p Π. Prf: Assume that Θ,P is crrect and that p Π. Extend Ip t a graded basis B fr P, i.e., a B fr which B Π j is a basis fr P Π j fr all j. Since Θ,P is crrect, it fllws that B must be linearly independent ver Θ. Since Ip = p n Θ, it fllws that (B\Ip) p is als linearly independent ver Θ, hence its span, Q say, is als crrect fr interplatin n Θ. If nw degip > degp =: j, then it fllws that dim Q Π j > dim P Π j, hence P is nt f minimal degree. Cnversely, if deg p deg Ip fr all plynmials p, then I(Π k ) Π k P, while, fr any crrect Θ,Q, I is 1-1 n Q, hence linear independence preserving, and ne therefre has dim(π k Q) =dimi(π k Q), while als I(Π k Q) Π k P. Therefre, P has the minimal degree prperty. Prperty (9) asserts that Π Θ Θ =Π Θ Π Θ. Fr its prf, bserve that Π Θ Θ =(exp Θ Θ ) =(exp Θ exp Θ ), that and that Π Θ Π Θ =(exp Θ ) (exp Θ ), (f g) =(f ) (g ), 22

24 hence that, fr any hmgeneus basis B fr Π Θ and any hmgeneus basis B fr Π Θ, the cllectin B B := {b b :(b, b ) B B } is a basis fr Π Θ Π Θ and in Π Θ Θ, hence a basis fr it since it cnsists f #(B B ) = #(Θ Θ )=dimπ Θ Θ elements. Fr i =1,...,d,letΘ i = {x i (0),...,x i (γ(i))} be a cllectin f γ(i) pintsinir. Inductive applicatin f (9) shws that (7.2) Π Θ = d Π γ(i) (IR) fr Θ = d i=1θ i. i=1 In cnjunctin with the mntnicity prperty (8), this implies that ur Π Θ cincides with the standard assignment even in the case when Θ is an rder-clsed subset f a rectangular grid i Θ i =: d i=1{x i (0),...,x i (γ(i))}. Here, we call Θ rder-clsed (r, a lwer set ) if it is f the frm Θ = {θ α : α Γ} fr θ α := (x 1 (α(1)),...,x d (α(d))), with Γ an rder-clsed subset f C γ, i.e., α Γ and β α implies β Γ, and C γ := {0,...,γ(1)} {0,...,γ(d)}. Thus, Γ= C α. α Γ We shw nw that the standard assignment fr this Θ, i.e., the space span{() β : β Γ}, des indeed cincide with Π Θ. Since, fr any α Γ, the subset Θ α := {θ β : β C α } f Θ is a cartesian prduct f sets frm IR, ur assignment fr it is necessarily Π α := span{() β : β α}, by (7.2). By (8), each such Π α must be cntained in Π Θ, hence span{() β : β Γ} Π Θ, and, since dim Π Θ =#Θ=#Γ,Π Θ must cincide with that span. Nte that the particular rdering f the pints x i (j) is arbitrary. Fr example, fr each f the fllwing three datapint distributins Θ x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x in the plane, the discussin just given prves that Π Θ =Π 1 (IR) Π 5 (IR) + Π 5 (IR) Π 1 (IR). x x x x x x x x x x x x x x x x x x x x Fr, each set is btained frm a rectangular set f 6 6 pints by retaining tw rws and tw clumns. It is nt all that hard t see, in each f these figures, the tw rectangles, ne 2 6 and the ther 6 2, which give rise t the crrespnding sum f plynmial spaces. Finally, (10) is prved in [5]. 23

25 8. Generalizatins With minr changes, the entire discussin can be extended t the situatin when we cnsider an arbitrary linearly independent sequence λ 1,λ 2,...,λ n f linear functinals instead f the particular linear functinals f f(ϑ) with ϑ in sme n-set Θ. Assuming the linear functinals λ i t be regular enugh t be representable as λ i f = f i,f fr f Π, with f i functins analytic at the rigin, the apprpriate Vandermnde-like matrix nw has in its ith rw the derivatives D α f i (0). Fr small enugh x, f i (x) =λ i (exp x ). The cmputatins are therwise unchanged. In particular, the hmgeneus plynmials g 1,...,g n cnstructed span a scale-invariant plynmial space f smallest degree n which the sequence λ 1,λ 2,...,λ n is maximally linearly independent. Further, g j,f (8.1) g j g j,g j j is the unique element in that space which agrees with f at the λ i. This general setting is discussed in [5] in detail, and als fr varius specific chices f the interplatin cnditins λ 1,λ 2,...,λ n. The discussin there cvers even linear functinals which are nt regular in the abve sense and als the situatin when we have infinitely many f them. Finally, we nte that, starting with an arbitrary basis f 1,f 2,...,f n fr sme linear subspace H f functins analytic at the rigin, the hmgeneus plynmials g 1,...,g n prvided by the algrithm frm a basis fr H = span{f : f H}. In particular, this leads t a cnstructin f a basis fr the space f plynmials in the span f the translates f a bx spline, as is detailed in [4]. References [1] C. de Br, B-frm basics, in Gemetric Mdeling, G. Farin ed., SIAM (1987), [2] C. de Br, Plynmial interplatin in several variables, Prceedings f the Cnference hnring Samuel D. Cnte, R. DeMill and J.R. Rice eds., Plenum Press, 199x, xxx xxx. [3] C. de Br and A. Rn, On multivariate plynmial interplatin, Cnstructive Apprximatin 6 (1990), [4] C. de Br and A. Rn, On ideals f finite cdimensin with applicatins t bx spline thery, J.Math.Anal.Appl. xx (199x), xxx xxx. [5] C. de Br and A. Rn, The least slutin fr the plynmial interplatin prblem, Cmp.Sci. Tech.Rep. #942, June, [6] MathWrks, MATLAB User s Guide, MathWrks Inc., Suth Natick MA, February