On the Lebesgue constant for the Xu interpolation formula

Transcription

1 O the Lebesgue costat for the Xu iterpolatio formula Le Bos Dept. of Mathematics ad Statistics, Uiversity of Calgary Caada Stefao De Marchi Dept. of Computer Sciece, Uiversity of Veroa Italy Marco Viaello Dept. of Pure ad Applied Mathematics, Uiversity of Padova Italy Correspodig author: Stefao De Marchi, Departmet of Computer Sciece, Uiversity of Veroa S.da Le Grazie 15, VERONA ITALY tel: ; fax:

2 Abstract I the paper [8], the author itroduced a set of Chebyshev-like poits for polyomial iterpolatio by a certai subspace of polyomials i the square [ 1, 1], ad derived a compact form of the correspodig Lagrage iterpolatio formula. I [1] we gave a efficiet implemetatio of the Xu iterpolatio formula ad we studied umerically its Lebesgue costat, givig evidece that it grows like Olog, beig the degree. The aim of the preset paper is to provide a aalytic proof that ideed the Lebesgue costat does have this order of growth.

3 1 Itroductio Suppose that K R d is a compact set with o-empty iterior. Let V be a subspace of Π d, the polyomials of degree i d variables, of dimesio dimv =: N. The give N poits X := {x k } N k=1 K, the polyomial iterpolatio problem associated to V ad X is the followig: for each f CK, the space of cotiuous fuctios o the compact K, fid a polyomial p V such that px k = fx k, k = 1,..., N. If this is always possible the problem is said to be uisolvet. Ad if this is ideed the case we may costruct the so-called Lagrage fudametal polyomials l j x with the property that l j x k = δ jk, the Kroecker delta. Further, the iterpolat itself may be writte as Lfx = N fx k l k x. k=1 The mappig f Lf may be regarded as a operator from CK equipped with the uiform orm to itself, ad as such has a operator orm L. Classically, whe K = [ 1, 1] ad V = Π 1, dimv = + 1, this orm is kow as the Lebesgue costat ad it is kow that the L C log ad that this miimal order of growth is attaied, for example, by the Chebyshev poits see e.g. []. I the multivariate case much less is kow. From Berma s Theorem cf. [6, Theorems 6.4 ad 6.5] it follows that for K = B d, the uit ball i R d, d, ad V = Π d, the Lebesgue costat has a miimal rate of growth of O d 1/. I the tesor product case, whe K = [ 1, 1] d ad V = d k=1 Π1, the L Clog d ad this miimal rate of growth is attaied for the tesor product of the uivariate Chebyshev poits. However, eve for the cube ad the polyomials of total degree, i.e., for K = [ 1, 1] d ad V = Π d, the miimal rate of growth is ot kow. Recetly Xu [7, 8] itroduced a set of Chebyshev-like poits for K = [ 1, 1], the square, ad he also provided a compact Lagrage iterpolatio formula based o these poits, establishig the coectio betwee the so-called miimal cubature formulas ad his Lagrage iterpolatio formula. We recall that a N-poit cubature formula of degree 1, has to satisfy N dimπ 1 +, 1 ad is called miimal whe the lower boud is attaied see the origial paper [4] by Möller. For eve, the miimal cubature formula correspodig to these Chebyshev-like poits was itroduced by Morrow ad Patterso i [5], ad later o exteded for odd by Xu i [8]. The coectio with Lagrage iterpolatio was studied by Xu i [8], by itroducig a certai subspace of polyomials, V = V, with the property that Π 1 V Π, 3

4 ad dimesio N = dimπ 1 + = + / for eve, N = dimπ = +1 / for odd. It should be remarked that V, although ot a total degree space of polyomials, is much closer to Π 1 tha to the correspodig tesor-product space k=1 Π1 1 which has dimesio. The umerical experimets of [1] gave us good evidece that the Lebesgue costat of Xu-like iterpolatio has growth of the order log just as i the tesor product case, ad i cotrast to the case of the ball where the miimal growth would be of order ad it is the purpose of this ote to prove that this is ideed the case. From this we may coclude that the poits studied by Xu i [7, 8], are excellet poits for practical polyomial iterpolatio. Moreover, our result also gives strog evidece that the miimal rate of growth for the Lebesgue costat for iterpolatio of polyomials of total degree o a square is of order log. This idicates a fudametal differece betwee a square ad a disk, where the miimal growth is of order, which perhaps surprig. We also remark that there has recetly bee itroduced a set of poits i the square, the so-called Padua poits cf. [3] for which V = Π, that are aother Chebyshev-like family, ad for which umerical experimets idicate that the Lebesgue costat has this miimal Olog growth. The Xu polyomial iterpolatio formula We start by recallig briefly the costructio of the Xu iterpolatio formula of degree o the square [ 1, 1]. I what follows we restrict, for simplicity s sake to eve degrees. Startig from the Chebyshev-Lobatto poits o the iterval [ 1, 1], that is z k = z k, = cos kπ, k = 0,...,, = m, the iterpolatio poits o the square studied by Xu, are defied as the two dimesioal array X N = {x r,s } of cardiality N = + /, x i,j+1 = z i, z j+1, 0 i m, 0 j m 1 3 x i+1,j = z i+1, z j, 0 i m 1, 0 j m. 4 The Xu iterpolat i Lagrage form of a give fuctio f o the square is where the polyomials K, x k,l are give by L Xu fx = fx k,l K x, x k,l K x k,l X x N k,l, x k,l, 5 K x, x k,l = 1 K +1x, x k,l + K x, x k,l 1 1k T x T y. 6 Here x, y are the coordiates of the geeric poit x ad T is the Chebyshev polyomial of the first kid of degree, T x = cos arccos x. 4

5 The polyomials K x, y ca be represeted i the form K x, y = D θ 1 + φ 1, θ + φ + D θ 1 + φ 1, θ φ + D θ 1 φ 1, θ + φ + D θ 1 φ 1, θ φ, 7 x = cos θ 1, cos θ, y = cos φ 1, cos φ, where the fuctio D is defied by D α, β = 1 cos 1/α cosα/ cos 1/β cos β/ cos α cos β. 8 As show i [8] the values K x k,l, x k,l are explicitly kow i terms of the degree, that is K x k,l, x k,l = / k = 0 or k =, l odd l = 0 or l =, k odd i all other cases. 9 Observe that this costructive approach yields immediately uisolvece of the iterpolatio problem, ce for ay give basis of V the correspodig Vadermode system has a solutio for every vector {fx k,l }, ad thus the Vadermode matrix is ivertible. 3 The Lebesgue costat of the Xu poits We will show that Theorem 1 The Lebesgue costat of the Xu poits Λ Xu, is bouded by Λ Xu 8 π log The proof will follow from a sequece of techical lemmas. Lemma 1 The fuctio D α, β ca be writte as D α, β = Proof. The proof is obtaied by simple trigoometric maipulatios. Ideed, ug the idetity cosa + B + cosa B cosa cosb =, we obtai 5

6 D α, β = 1 cosα + cos 1α [cosβ cos 1β] 4 cos α cos β = 1 cosα cosβ + [cos 1α cos 1β]. 4 cos α cos β The, by the fact that cosa cosb = A+B A B, the result follows. Now, for the poits x = cos θ 1, cos θ ad x k,l = cos φ 1, cos φ, by ug 6 ad 7, we have K x, x k,l = 1 {D + D +1 θ 1 + φ 1, θ + φ + D + D +1 θ 1 + φ 1, θ φ + D + D +1 θ 1 φ 1, θ + φ + D + D +1 θ 1 φ 1, θ φ } 1 1k cosθ 1 cosθ 1 Sice we wat to boud Λ Xu, we start by fidig a upper boud for Kx, x k,l. First we observe that from Lemma 1 D α, β + D +1 α, β Lemma Proof. Let θ = ad φ =, the by ug simple trigoometric idetities, the umerator ca be re-writte as Thus θ cos θ + θ cos θ φ cos φ + φ cos φ + θ cos θ θ cos θ φ cos φ φ cos φ + 1θ + 1φ 1θ 1φ + θ φ θ φ = θ φ θ φ cos θ cos φ cos θ cos φ + θ φ θ φ θ φ θ φ +. By Lemma, the followig upper boud for D + D +1 α, β holds. 6

7 Lemma 3 D α, β + D +1 α, β = Now we cosider Kx, x k,l. Lettig x = cos θ 1, cos θ ad x k,l = cos φ 1, cos φ, we kow that Kx, x k,l ca be writte as i 1. Thus, from Lemmas 1 ad, K x, x k,l 1 + θ1 +θ +φ 1 +φ θ1 +φ 1 θ φ θ1 +θ +φ 1 +φ θ1 +φ 1 θ φ + 3 other terms + 14 ad so for the Lagrage polyomials L k,l x = K x, x k,l Kx k,l, x k,l K x, x k,l, k = 0,...,, l = 0,..., The Xu poits are of two types cf. 3 ad 4 that for short we call typea ad typeb, that is typea : x i,j+1 = z i, z j+1 ad typeb : x i+1,j = z i+1, z j, where z s are as i ad i = 0,...,, j = 0,..., 1. Cosider the sum of the Lagrage polyomials for the poits of typea. I the boud of K x, x k,l see above formula 14, there are four terms plus a costat that sum up to 1 which does ot cotribute to the domiat growth of the Lebesgue costat. Hece, we eed oly to boud the four terms ivolvig the es. Ideed, A typea := / i=0 / 1 j=0 / i=0 / i=0 L i,j+1 16 / 1 j=0 / 1 j=0 K x, x i,j+1 = / + 1 θ1 +θ +φ 1 +φ θ1 +φ 1 θ φ θ1 +θ +φ 1 +φ θ1 +φ 1 θ φ + 3 other terms / 1 θ1 +θ +φ 1 +φ θ1 +φ 1 θ φ i=0 j=0 θ1 +θ +φ 1 +φ θ1 +φ 1 θ φ

8 Let A be the first of these four terms. Sice x i,j+1 = as A = 1 / i=0 / 1 j=0 θ1 +θ +i+j+1π/ θ1 +θ +i+j+1π/ cos iπ j+1π, cos θ1 θ +i j 1π/ θ1 θ +i j 1π/, we ca write it. 17 Now, chage variables i the double sum. Lettig k = i + j ad m = i j. This is a 1-1 mappig betwee the pairs of itegers i, j, 0 i, 0 j 1 i a subset of the pairs of itegers k, m, 0 k 1, m. Hece, A 1 1 = 1 1 = 1 / θ1 +θ +k+1π/ k=0 m= / θ1 +θ +k+1π/ θ1 +θ +k+1π/ / k=0 θ1 +θ +k+1π/ m= / 1 θ1 +θ +k+1π/ k=0 θ1 +θ +k+1π/ 1 / m= / θ1 θ +m 1π/ θ1 θ +m 1π/ θ1 θ +m 1π/ θ1 θ +m 1π/ θ1 θ +m 1π/ θ1 θ +m 1π/ The ext step cosists i boudig each factor separately. Start with the first i Lemma 4 Suppose that φ [ π, π], ad set θ k = φ + k+1π. The, 1 1 k=0 θ k θ k log π Proof. Let 0 φ 0 < φ 1 < < φ 1 π be the set of agles {θ k } 1 k=0 The, the φ j are equally spaced, i.e., take modulo π. The, ce θ ± π = θ, we have 1 1 k=0 θ k θ k φ j φ j 1 = π, j = 1,..., 1. = 1 1 φ k φ k k= φ k ce each φ k term 1 =1 k= φ k π csc φ k π π = 4 + π k= π π/ π/ π/ π/ k= cscθ dθ by the covexity of csc θ cscθdθ = 4 + { log csc θ + cot θ π 8 π π }

9 = 4 + π log csc π + cot π 4 + π log csc π 4 + π log 1 π π = 4 + π log. For the secod factor i 18 we have a similar result. Lemma 5 For eve ad φ [ π, π], set θ k = φ + k+1π. The, 1 / k= / θ k θ k log π Proof. The argumet is the same as the previous Lemma, except there is oe more term i the sum. But this term like all the others is bouded by 1.. Proof of the Mai Theorem. It follows that A typea is bouded by plus four times the boud for A, i.e., A typea + 4 π log + 5. The, icludig the same boud for the typeb poits, we have { } Λ Xu + 4 π log + 5 = 8 π log Ackowledgmets. First of all, we wish to thak Yua Xu for his suggestios i the derivatio of the proof of the mai theorem. This work has bee supported by the research project CPDA0891 Efficiet approximatio methods for olocal discrete trasforms of the Uiversity of Padova, the ex-60% fuds of the Uiversity of Veroa, ad by the GNCS-INdAM. Refereces [1] L. Bos, M. Caliari, S. De Marchi ad M. Viaello, A umerical study of the Xu polyomial iterpolatio formula i two variables, i press o Computig 005; preprit available at marcov/publicatios.html. [] L. Brutma, Lebesgue fuctios for polyomial iterpolatio - a survey, A. Numer. Math , [3] M. Caliari, S. De Marchi ad M. Viaello, Bivariate polyomial iterpolatio o the square at ew odal sets, Appl. Math. Comput ,

10 [4] H. M. Möller, Kubaturformel mit miimaler Kotezahl, Numer. Math /76, [5] C. R. Morrow ad T. N. L. Patterso: Costructio of algebraic cubature rules ug polyomial ideal theory. SIAM J. Numer. Aal., 155, [6] M. Reimer, Multivariate Polyomial Approximatio, Iteratioal Series of Numerical Mathematics, Vol. 144, Birkhäuser, 003. [7] Y. Xu, Commo zeros of polyomials i several variables i higher dimesioal quadratures, Pitma Res. Notes i Math. Series, Logma, Essex [8] Y. Xu, Lagrage iterpolatio o Chebyshev poits of two variables, J. Approx. Theory ,