International audienceThe so-called Bernstein operators were introduced by S.N. Bernstein in 1912 to give a constructive proof of Weierstrass' theorem. We show how to extend his result to Müntz spaces on positive intervals

Ait-Haddou, Rachid

Mazure, Marie-Laurence

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 351 (2013) 849–852Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comNumerical analysisApproximation by Müntz spaces on positive intervalsApproximation par espaces de Müntz sur un intervalle positifRachid Ait-Haddou a, Marie-Laurence Mazure ba Geometric Modeling and Scientific Visualization Center, King Abdullah University of Science and Technology, Saudi Arabiab Laboratoire Jean-Kuntzmann, Université Joseph-Fourier, BP 53, 38041 Grenoble cedex 9, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 6 September 2013Accepted 18 October 2013Available online 12 November 2013Presented by Philippe G. CiarletThe so-called Bernstein operators were introduced by S.N. Bernstein in 1912 to give aconstructive proof of Weierstrass’ theorem. We show how to extend his result to Müntzspaces on positive intervals.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éEn 1912, les opérateurs dits de Bernstein permirent à S.N. Bernstein de donner une preuveconstructive du théorème de Weierstrass. Nous étendons ce résultat aux espaces de Müntzsur des intervalles positifs.© 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionThe famous Bernstein operator Bk of degree k on a given non-trivial interval [a,b], associates with any F ∈ C0([a,b]) thepolynomial function:Bk F (x) :=k∑i=0F((1 − ik)a + ikb)Bki , x ∈ [a,b], (1)where (Bk0, . . . , Bkk) is the Bernstein basis of degree k on [a,b], i.e., Bki (x) :=(ki)( x−ab−a )i( b−xb−a )k−i . It reproduces any affinefunction U on [a,b], in the sense that BkU = U . In [5], S.N. Bernstein proved that, for every function F ∈ C0([a,b]),limk→+∞ ‖F −Bk F‖∞ = 0. In Section 3, we show how this result extends to the class of Müntz spaces (i.e., spaces spannedby power functions) on a given positive interval [a,b], see Theorem 3.1. Beforehand, in Section 2, we briefly remind thereader how to define operators of the Bernstein-type in Extended Chebyshev spaces.2. Extended Chebyshev spaces and Bernstein operatorsThroughout this section, [a,b] is a fixed non-trivial real interval. For any n  0, a given (n + 1)-dimensional space E ⊂Cn([a,b]) is said to be an Extended Chebyshev space (for short, EC-space) on [a,b] when any non-zero element of E vanishesat most n times on [a,b] counting multiplicities up to (n + 1).E-mail addresses: Rachid.AitHaddou@kaust.edu.sa (R. Ait-Haddou), mazure@imag.fr (M.-L. Mazure).1631-073X/$ – see front matter © 2013 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2013.10.018850 R. Ait-Haddou, M.-L. Mazure / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 849–852Let E be an (n + 1)-dimensional EC-space on [a,b]. Then, E possesses bases (B0, . . . , Bn) such that, for i = 0, . . . ,n, Bivanishes exactly i times at a and (n − i) times at b and is positive on ]a,b[. We say that such a basis is the Bernstein basisrelative to (a,b) if it additionally satisfies∑ni=0 Bi = 1, where 1 is the constant function 1(x) = 1, x ∈ [a,b]. Let us recall thatE possesses a Bernstein basis relative to (a,b) if and only if, firstly it contains constants, and secondly the n-dimensionalspace DE := {D F := F ′ | F ∈ E} is an EC-space on [a,b]. Note that the second property is not an automatic consequence ofthe first one, see [8] and other references therein.As an instance, given any pairwise distinct λ0, . . . , λk , the so-called Müntz space M(λ0, . . . , λk), spanned over a givenpositive interval [a,b] (i.e., a > 0) by the power functions xλi , 0  i  k, is a (k + 1)-dimensional EC-space on [a,b]. If λ0 = 0,since D(M(λ0, . . . , λk)) = M(λ1 − 1, . . . , λk − 1), the space M(λ0, . . . , λk) possesses a Bernstein basis relative to (a,b).For the rest of the section, we assume that E ⊂ Cn([a,b]) contains constants and that DE is an (n-dimensional) EC-spaceon [a,b]. We denote by (B0, . . . , Bn) the Bernstein basis relative to (a,b) in E.Definition 2.1. A linear operator B : C0([a,b]) → E is said to be a Bernstein operator based on E when, firstly it is of the form:BF :=k∑i=0F (ζi)Bi, for some a = ζ0 < ζ1 < · · · < ζn = b, (2)and secondly it reproduces a two-dimensional EC-space U on [a,b], in the sense that BV = V for all V ∈U.Any Bernstein operator B is positive (i.e., F  0 implies BF  0) and shape preserving due to the properties of Bernsteinbases in EC-spaces, see [8]. Note that everything concerning Bernstein-type operators in EC-spaces with no Bernstein basescan be deduced from Bernstein operators as defined above [8,9].Theorem 2.2. Given n  2, let E ⊂ Cn([a,b]) contain constants. Assume that DE is an n-dimensional EC-space on [a,b]. For a functionU ∈ E, expanded in the Bernstein basis relative to (a,b) as U := ∑ni=0 ui Bi , the following properties are equivalent:(i) u0, . . . , un form a strictly monotonic sequence;(ii) there exists a nested sequence E1 ⊂ E2 ⊂ · · · ⊂ En−1 ⊂ En := E, where E1 := span(1, U ) and where, for i = 1, . . . ,n − 1, Ei isan (i + 1)-dimensional EC-space on [a,b];(iii) there exists a Bernstein operator based on E which reproduces U .In [8] it was proved that there exists a one-to-one correspondence between the set of all Bernstein operators based on Eand the set of all two-dimensional EC-spaces U they reproduce. In particular, if (i) holds, then the unique Bernstein operatorbased on E reproducing U is defined by (2) with:ζi := U−1(ui), 0  i  n. (3)Note that this is meaningful because (i) implies the strict monotonicity of U on [a,b]. Condition (ii) of Theorem 2.2 yieldsthe following corollary.Corollary 2.3. Given an integer n  1, consider a nested sequence:En ⊂ En+1 ⊂ · · · ⊂ Ep ⊂ Ep+1 ⊂ · · · , (4)where En contains constants and for any p  n, DEp is a p-dimensional EC-space on [a,b]. Let U ∈ En be a non-constant functionreproduced by a Bernstein operator Bn based on En. Then, U is also reproduced by a Bernstein operator Bp based on Ep for any p > n.Remark 2.4. In the situation described in Corollary 2.3, a natural question arises: given F ∈ C0([a,b]), does the sequenceBk F , k  n, converges to F in C0([a,b]) equipped with the infinite norm? Obviously, for this to be true for any F ∈ C0([a,b]),it is necessary that⋃kn Ek be dense in C0([a,b]). The example of Müntz spaces proves that this is not always satisfied.3. Müntz spaces over positive intervalsThroughout this section, we consider a fixed positive interval [a,b], a fixed infinite sequence of real numbers λk , k  0,assumed to satisfy:0 = λ0 < λ1 < · · · < λk < λk+1 < · · · , limk→+∞λk = +∞. (5)We are concerned with the corresponding nested sequence of Müntz spaces:M(λ0) ⊂ M(λ0, λ1) ⊂ · · · ⊂ M(λ0, . . . , λk) ⊂ M(λ0, . . . , λk, λk+1) ⊂ · · · (6)R. Ait-Haddou, M.-L. Mazure / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 849–852 851Given any n  1, for each k  n, we can select a Bernstein operator Bk based on M(λ0, . . . , λk). Assume the sequence Bk ,k  n, to satisfy:limk→+∞‖F −Bk F‖∞ = 0 for any F ∈ C0([a,b]). (7)Then, the union of all spaces M(λ0, . . . , λk), k  0, is dense in C0([a,b]) equipped with the infinite norm. As is well known,this holds if and only if the sequence (5) fulfils the so-called Müntz density condition below [4,6]:∑i11λi= +∞. (8)As an instance, the Müntz condition (8) is satisfied when λk =  + 1 for all k  1. This case was addressed in [8]. Con-vergence – in the sense of (7) – was proved there under the assumption that each Bk reproduced the function xλ1 . Thisconvergence result includes the classical Bernstein operators [5] obtained with  = 0. Below we extend it to the generalinteresting situation of sequences of Müntz Bernstein operators Bk all reproducing the same two-dimensional EC-space (seeRemark 2.4).Theorem 3.1. Given n  1, let E1 ⊂ M(λ0, . . . , λn) be a two-dimensional EC-space reproduced by a Bernstein operator Bk based onM(λ0, . . . , λk) for any k  n. Then, if the Müntz density condition (8) holds, the sequence Bk, k  n, converges in the sense of (7).Before starting the proof, let us introduce some notations. For k  1, denote by (Bk,0, . . . , Bk,k) the Bernstein basis relativeto (a,b) in the Müntz space M(λ0, . . . , λk). We consider the functions:U∗(x) = xλ1 , V p(x) := xλp , p  2, x ∈ [a,b],expanded in the successive Bernstein bases as:U∗ =k∑i=0u∗k,i Bk,i for all k  1, V p =k∑i=0v p,k,i Bk,i for all k  p. (9)With these notations, the key-point to prove Theorem 3.1 is the following lemma, for the proof of which we refer to [1],see also [2].Lemma 3.2. Assume that the Müntz density condition (8) holds. Then, we have:limk→+∞max0ik∣∣(u∗k,i)λpλ1 − v p,k,i∣∣ = 0 for all p  2. (10)Proof of Theorem 3.1. • Let us start with the simplest example n = 1. Then, E1 = span(1, U∗). For each k  1, the uniqueoperator B∗k which reproduces E1 is given by:B∗k F :=k∑i=0F(ζ ∗k,i)Bk,i, with, for i = 0, . . . ,k, ζ ∗k,i :=(u∗k,i) 1λ1 . (11)According to Korovkin’s theorem for positive linear operators [7], we just have to select a function F so that 1, U ∗ , F spana three-dimensional EC-space on [a,b] and prove that limk→+∞ ‖F − B∗k F‖∞ = 0 for this specific F . We can thus choosefor instance F := V 2. Actually, we will more generally prove the result with F = V p , for any p  2. Using (9) and (11), weobtain, for any k  p,∥∥B∗k V p − V p∥∥∞ =∥∥∥∥∥k∑i=0(V p(ζ ∗k,i) − v p,k,i)Bk,i∥∥∥∥∥∞ max0ik∣∣V p(ζ ∗k,i) − v p,k,i∣∣. (12)On account of (11), Lemma 3.2 yields the expected result:limk→+∞∥∥B∗k V p − V p∥∥∞ = 0 for each p  2.• We now assume that n > 1. Select a strictly increasing function U ∈ E1. Condition (ii) of Theorem 2.2 enables us to select afunction V ∈ M(λ0, . . . , λn) so that the functions 1, U , V span a three-dimensional EC-space on [a,b]. For any k  n, expandU , V as:852 R. Ait-Haddou, M.-L. Mazure / C. R. Acad. Sci. Paris, Ser. I 351 (2013) 849–852U =k∑i=0uk,i Bk,i, V =k∑i=0vk,i Bk,i .We know that, for each k  n, the sequence (uk,0, . . . , uk,k) is strictly increasing, and that the Bernstein operator Bk isdefined by formula (2) with ζk,i := U−1(uk,i) for i = 0, . . . ,k. Via expansions of U and V in the basis (1, U∗, V 2, . . . , Vn) ofthe Müntz space M(λ0, . . . , λn), Lemma 3.2 readily proves that:limk→+∞max0ik∣∣U(ζ ∗k,i) − uk,i∣∣ = 0 = limk→+∞ max0ik∣∣V (ζ ∗k,i) − vk,i∣∣. (13)The left part in (13) can be written as limk→+∞ max0ik |U (ζ ∗k,i) − U (ζk,i)| = 0. On this account, the uniform continuityof the function V ◦ U−1 and the right part in (13) prove that limk→+∞ max0ik |V (ζk,i) − vk,i | = 0, thus implying thatlimk→+∞ ‖Bk V − V ‖∞ = 0. By Korovkin’s theorem, (7) is satisfied. Remark 3.3. Given n  2, one can apply Theorem 3.1 with E1 := span(1, Vn) = M(λ0, λn), due to the nested sequence ofMüntz spaces M(λ0, λ1, . . . , λi−1, λn) for 1  i  n. Note that Theorem 3.1 contains in particular the Bernstein-type resultexpected in [3].References[1] R. Ait-Haddou, Dimension elevation in Müntz spaces: a new emergence of the Müntz condition, in press, arXiv:1309.0938.[2] R. Ait-Haddou, Y. Sakane, T. Nomura, A Müntz type theorem for a family of corner cutting schemes, Comput. Aided Geom. Design 30 (2013) 240–253.[3] J.M. Aldaz, O. Kounchev, H. Render, Bernstein operators for exponential polynomials, Constr. Approx. 20 (2009) 345–367.[4] J.M. Almira, Müntz type theorems I, Surv. Approx. Theory 3 (2007) 152–194.[5] S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Soc. Math. Kharkov 13 (1912) 1–2.[6] P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995.[7] P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Corp., 1960.[8] M.-L. Mazure, Bernstein-type operators in Chebyshev spaces, Numer. Algorithms 52 (2009) 93–128.[9] M.-L. Mazure, Finding all systems of weight functions associated with a given Extended Chebyshev space, J. Approx. Theory 163 (2011) 363–376.

Approximation by Müntz spaces on positive intervals

Rachid Ait-Haddou

Marie-Laurence Mazure

Comptes Rendus Mathématique

International audienceThe so-called Bernstein operators were introduced by S.N. Bernstein in 1912 to give a constructive proof of Weierstrass' theorem. We show how to extend his result to Müntz spaces on positive intervals

Hal - Université Grenoble Alpes

Approximation by Mu¨ntz spaces on positive intervalsRachid Ait-Haddou, Marie-Laurence MazureTo cite this version:Rachid Ait-Haddou, Marie-Laurence Mazure. Approximation by Mu¨ntz spaces on positiveintervals. Comptes Rendus Mathe´matique, Elsevier Masson, 2013, 351 (21-22), pp.849-852.<10.1016/j.crma.2013.10.018>. <hal-00983519>HAL Id: hal-00983519https://hal.archives-ouvertes.fr/hal-00983519Submitted on 25 Apr 2014HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Numerical AnalysisApproximation by Mu¨ntz spaces on positive intervalsRachid Ait-Haddou a, Marie-Laurence Mazure b,aGeometric Modeling and Scientific Visualization Center, KAUST, Saudia-ArabiabLaboratoire Jean Kuntzmann Universite´ Joseph Fourier, BP 53, 38041 Grenoble cedex 9, FranceAbstractThe so-called Bernstein operators were introduced by S.N. Bernstein in 1912 to give a constructive proof ofWeierstrass’ theorem. We show how to extend his result to Mu¨ntz spaces on positive intervals. To cite this article:R. Ait-Haddou and M.-L. Mazure, C. R. Acad. Sci. Paris, Ser. ???? (200?).Approximation par espaces de Mu¨ntz sur un intervalle positif. Re´sume´En 1912, les ope´rateurs dits de Bernstein permirent a` S.N. Bernstein de donner une preuve constructive duthe´ore`me de Weierstrass. Nous e´tendons ce re´sultat aux espaces de Mu¨ntz sur des intervalles positifs. Pour citercet article : R. Ait-Haddou and M.-L. Mazure, C. R. Acad. Sci. Paris, Ser. ? ? ? ? (200 ?).1. IntroductionThe famous Bernstein operator Bk of degree k on a given non-trivial interval [a, b], associates with anyF ∈ C0([a, b]) the polynomial functionBkF (x) :=k∑i=0F((1−ik)a+ikb)Bki , x ∈ [a, b], (1)where (Bk0 , . . . , Bkk ) is the Bernstein basis of degree k on [a, b], i.e., Bki (x) := (ki)(x−ab−a)i( b−xb−a)k−i. Itreproduces any affine function U on [a, b], in the sense that BkU = U . In [5], S.N. Bernstein proved that,for all function F ∈ C0([a, b]), limk→+∞ ‖F − BkF‖∞ = 0. In Section 3 we show how this result extendsto the class of Mu¨ntz spaces ( i.e., spaces spanned by power functions) on a given positive interval [a, b],see Theorem 3.1. Beforehand, in Section 2 we briefly remind the reader how to define operators of theBernstein-type in Extended Chebyshev spaces.2. Extended Chebyshev spaces and Bernstein operatorsThroughout this section, [a, b] is a fixed non-trivial real interval. For any n > 0, a given (n + 1)-dimensional space E ⊂ Cn([a, b]) is said to be an Extended Chebyshev space (for short, EC-space) on [a, b]when any non-zero element of E vanishes at most n times on [a, b] counting multiplicities up to (n+ 1).Email addresses: Rachid.AitHaddou@kaust.edu.sa (Rachid Ait-Haddou), mazure@imag.fr (Marie-Laurence Mazure).Preprint submitted to Elsevier Science October 21, 2013Let E be an (n+1)-dimensional EC-space on [a, b]. Then, E possesses bases (B0, . . . , Bn) such that, fori = 0, . . . , n, Bi vanishes exactly i times at a and (n− i) times at b and is positive on ]a, b[. We say thatsuch a basis is the Bernstein basis relative to (a, b) if it additionally satisfies∑ni=0 Bi = 1I, where 1I is theconstant function 1I(x) = 1, x ∈ [a, b]. Let us recall that E possesses a Bernstein basis relative to (a, b) ifand only if, firstly it contains constants, and secondly the n-dimensional space DE := {DF := F ′ |F ∈ E}is an EC-space on [a, b]. Note that the second property is not an automatic consequence of the first one,see [8] and other references therein.As an instance, given any pairwise distinct λ0, . . . , λk, the so-calledMu¨ntz spaceM(λ0, . . . , λk), spannedover a given positive interval [a, b] ( i.e., a > 0) by the power functions xλi , 0 6 i 6 k, is a (k + 1)-dimensional EC-space on [a, b]. If λ0 = 0, since D (M(λ0, . . . , λk)) = M(λ1−1, . . . , λk−1), the spaceM(λ0, . . . , λk) possesses a Bernstein basis relative to (a, b).For the rest of the section we assume that E ⊂ Cn([a, b]) contains constants and that DE is an (n-dimensional) EC-space on [a, b]. We denote by (B0, . . . , Bn) the Bernstein basis relative to (a, b) in E.Definition 2.1 A linear operator B : C0([a, b])→ E is said to be a Bernstein operator based on E when,firstly it is of the formBF :=k∑i=0F(ζi)Bi, for some a = ζ0 < ζ1 < · · · < ζn = b, (2)and secondly it reproduces a two-dimensional EC-space U on [a, b], in the sense that BV = V for allV ∈ U.Any Bernstein operator B is positive ( i.e., F > 0 implies BF > 0) and shape preserving due to theproperties of Bernstein bases in EC-spaces, see [8]. Everything concerning Bernstein-type operators inEC-spaces with no Bernstein bases can be deduced from Bernstein operators as defined above [8], [9].Theorem 2.2 Given n > 2, let E ⊂ Cn([a, b]) contain constants. Assume that DE is an n-dimensionalEC-space on [a, b]. For a function U ∈ E, expanded in the Bernstein basis relative to (a, b) as U :=∑ni=0 uiBi, the following properties are equivalent:(i) u0, . . . , un form a strictly monotonic sequence;(ii) there exists a nested sequence E1 ⊂ E2 ⊂ · · · ⊂ En−1 ⊂ En := E, where E1 := span(1I, U) and where,for i = 1, . . . , n− 1, Ei is an (i+ 1)-dimensional EC-space on [a, b];(iii) there exists a Bernstein operator based on E which reproduces U .In [8] it was proved that there exists a one-to-one correspondence between the set of all Bernsteinoperator based on E and the set of all two-dimensional EC-spaces U they reproduce. In particular, if (i)holds, then the unique Bernstein operator based on E reproducing U is defined by (2) withζi := U−1(ui), 0 6 i 6 n. (3)Note that this is meaningful since (i) implies the strict monotonicity of U on [a, b]. Condition (ii) of Th.2.2 yields the following corollary.Corollary 2.3 Given an integer n > 1, consider a nested sequenceEn ⊂ En+1 ⊂ · · · ⊂ Ep ⊂ Ep+1 ⊂ · · · , (4)where En contains constants and for any p > n, DEp is a p-dimensional EC-space on [a, b]. Let U ∈ En bea non-constant function reproduced by a Bernstein operator Bn based on En. Then, U is also reproducedby a Bernstein operator Bp based on Ep for any p > n.Remark 2.4 In the situation described in Corollary 2.3, a natural question arises: given F ∈ C0([a, b]),does the sequence BkF , k > n, converges to F in C0([a, b]) equipped with the infinite norm? Obviously,2for this to be true for any F ∈ C0([a, b]), it is necessary that ∪k>nEk be dense in C0([a, b]). The exampleof Mu¨ntz spaces proves that this is not always satisfied.3. Mu¨ntz spaces over positive intervalsThroughout this section we consider a fixed positive interval [a, b], a fixed infinite sequence of realnumbers λk, k > 0, assumed to satisfy0 = λ0 < λ1 < · · · < λk < λk+1 < · · · , limk→+∞λk = +∞. (5)We are interested with the corresponding nested sequence of Mu¨ntz spacesM(λ0) ⊂M(λ0, λ1) ⊂ · · · ⊂M(λ0, . . . , λk) ⊂M(λ0, . . . , λk, λk+1) ⊂ · · · (6)Given any n > 1, for each k > n, we can select a Bernstein operator Bk based on M(λ0, . . . , λk). Assumethe sequence Bk, k > n, to satisfylimk→+∞‖F − BkF‖∞ = 0 for any F ∈ C0([a, b]). (7)Then, the union of all spaces M(λ0, . . . , λk), k > 0, is dense in C0([a, b]) equipped with the infinite norm.As is well-known, this holds if and only if the sequence (5) fulfils the so-called Mu¨ntz density conditionbelow [4], [6], ∑i>11λi= +∞. (8)As an instance, the Mu¨ntz condition (8) is satisfied when λk = ℓ+1 for all k > 1. This case was addressedin [8]. Convergence – in the sense of (7)– was proved there under the assumption that each Bk reproducedthe function xλ1 . This convergence result includes the classical Bernstein operators [5] obtained withℓ = 0. Below we extend it to the general interesting situation of sequences of Mu¨ntz Bernstein operatorsBk all reproducing the same two-dimensional EC-space (see Remark 2.4).Theorem 3.1 Given n > 1, let E1 ⊂ M(λ0, . . . , λn) be a two-dimensional EC-space reproduced by aBernstein operator Bk based on M(λ0, . . . , λk) for any k > n. Then, if the Mu¨ntz density condition (8)holds, the sequence Bk, k > n, converges in the sense of (7).Before starting the proof, let us introduce some notations. For k > 1, denote by (Bk,0, . . . , Bk,k) theBernstein basis relative to (a, b) in the Mu¨ntz space M(λ0, . . . , λk). We consider the functionsU∗(x) = xλ1 , Vp(x) := xλp , p > 2, x ∈ [a, b],expanded in the successive Bernstein bases asU∗ =k∑i=0u∗k,iBk,i for all k > 1, Vp =k∑i=0vp,k,iBk,i for all k > p. (9)With these notations, the key-point to prove Theorem 3.1 is the following lemma, for the proof of whichwe refer to [2], see also [1].Lemma 3.2 Assume that the Mu¨ntz density condition (8) holds. Then, we havelimk→+∞max06i6k∣∣∣∣(u∗k,i)λpλ1 − vp,k,i∣∣∣∣ = 0 for all p > 2. (10)Proof of Theorem 3.1: • Let us start with the simplest example n = 1. Then, E1 = span(1I, U∗). For eachk > 1, the unique operator B∗k which reproduces E1 is given byB∗kF :=k∑i=0F(ζ∗k,i)Bk,i, with, for i = 0, . . . , k, ζ∗k,i :=(u∗k,i) 1λ1 (11)3According to Korovkin’s theorem for positive linear operators [7], we just have to select a function F sothat 1I, U∗, F span a three-dimensional EC-space on [a, b] and prove that limk→+∞ ‖F − B∗kF‖∞ = 0 forthis specific F . We can thus choose for instance F := V2. Actually we will more generally prove the resultwith F = Vp, for any p > 2. Using (9) and (11), we obtain, for any k > p,‖B∗kVp − Vp‖∞ =∥∥∥∥∥k∑i=0(Vp(ζ∗k,i)− vp,k,i)Bk,i∥∥∥∥∥∞6 max06i6k∣∣Vp(ζ∗k,i)− vp,k,i∣∣ . (12)On account of (11), Lemma 3.2 yields the expected resultlimk→+∞‖B∗kVp − Vp‖∞ = 0 for each p > 2.• We now assume that n > 1. Select a strictly increasing function U ∈ E1. Condition (ii) of Theorem 2.2enables us to select a function V ∈M(λ0, . . . , λn) so that the functions 1I, U, V span a three-dimensionalEC-space on [a, b]. For any k > n, expand U, V asU =k∑i=0uk,iBk,i, V =k∑i=0vk,iBk,i.We know that, for each k > n, the sequence (uk,0, . . . , uk,k) is strictly increasing, and that the Bernsteinoperator Bk is defined by formula (2) with ζk,i := U−1 (uk,i) for i = 0, . . . , k. Via expansions of U and Vin the basis (1I, U∗, V2, . . . , Vn) of the Mu¨ntz space M(λ0, . . . , λn), Lemma 3.2 readily proves thatlimk→+∞max06i6k|U(ζ∗k,i)− uk,i| = 0 = limk→+∞max06i6k|V (ζ∗k,i)− vk,i| (13)The left part in (13) can be written as limk→+∞max06i6k |U(ζ∗k,i)−U(ζk,i)| = 0. On this account, the uni-form continuity of the function V ◦U−1 and the right part in (13) prove that limk→+∞max06i6k |V (ζk,i)−vk,i| = 0, thus implying that limk→+∞ ‖BkV − V ‖∞ = 0. By Korovkin’s theorem, (7) is satisfied. ✷Remark 3.3 Given n > 2, one can apply Theorem 3.1 with E1 := span(1I, Vn) = M(λ0, λn), due to thenested sequence of Mu¨ntz spaces M(λ0, λ1, . . . , λi−1, λn) for 1 6 i 6 n. Note that Theorem 3.1 containsin particular the Bernstein-type result expected in [3].References[1] R. Ait-Haddou, Y. Sakane, and T. Nomura, A Mu¨ntz type theorem for a family of corner cutting schemes, Comp.Aided Geom. Design, 30 (2013), 240–253.[2] R. Ait-Haddou, Dimension elevation in Mu¨ntz spaces: a new emergence of the Mu¨ntz condition, to appear,arXiv:1309.0938.[3] J.M. Aldaz, O. Kounchev, and H. Render, Bernstein Operators for Exponential Polynomials, Constr. Approx.,20 (2009), 345–367.[4] J. M. Almira, Mu¨ntz type theorems I, Surveys Approx. Theory, 3 (2007), 152–194.[5] S.N. Bernstein, De´monstration du the´ore`me de Weierstrass fonde´e sur le calcul des probabilite´s, Comm. Soc. Math.Kharkov, 13 (1912), 1-2.[6] P. Borwein and T. Erde´lyi, Polynomials and Polynomial Inequalities, Springer-Verlag, Graduate Texts inMathematics, Vol. 161, New York, 1995.[7] P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Corp., 1960.[8] M.-L. Mazure, Bernstein-type operators in Chebyshev spaces, Num. Algorithms, 52 (2009), 93–128.[9] M.-L. Mazure, Finding all systems of weight functions associated with a given Extended Chebyshev space, J. Approx.Theory, 163 (2011), 363–376.4

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Approximation by Müntz spaces on positive intervals

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