Today information is managed within increasingly complicated Web applications which often rely on similar information models. Finding a reusable and sufficiently generic information model for managing resources and their metadata would greatly simplify development of Web applications. This article presents such an information model, namely the Resource and Metadata Management Model (ReM3). The information model builds upon Web architecture and standards, more specifically the Linked Data principles when managing resources together with their metadata. It allows to express relations between metadata and to keep track of provenance and access control. In addition to this information model, the architecture of the reference implementation is described along with a Web application that builds upon it. To show the taken approach in practice, several real-world examples are presented as showcases. The information model and its reference implementation have been evaluated from several perspectives, such as the suitability for resource annotation, a preliminary scalability analysis and the adoption in a number of projects. This evaluation in various complementary dimensions shows that ReM3 has been successfully applied in practice and can be considered a serious alternative when developing Web applications where resources are managed along with their metadata.QC 20140417</p

Nagy, Béla

Totik, Vilmos

Ebner, Hannes

Palmér, Matthias

Academic Archive On-line (KTH, Sweden)

An information model for managing resources and their metadata

Publikationer från KTH

Digitala Vetenskapliga Arkivet - Academic Archive On-line

Béla Nagy

Vilmos Totik

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Constructive Approximation

Bernstein’s Inequality for Algebraic Polynomials on Circular Arcs

Nagy Béla

Totik Vilmos

SZTE Publicatio Repozitórium - SZTE - Repository of Publications

Bernstein’s inequality for algebraic polynomialson circular arcs∗Be´la Nagy and Vilmos Totik†October 22, 2011AbstractIn this paper we prove a sharp Bernstein inequality for algebraic poly-nomials on circular arcs.1 ResultsInequalities for algebraic or trigonometric polynomials play a fundamental rolein various problems ranging from number theory to differential equations. Oneof the most classical one is Bernstein’s inequality: if Pn is a polynomial of degreeat most n, C1 denotes the unit circle and ‖ · ‖K denotes supremum norm on aset K then|P ′n(z)| ≤ n‖Pn‖C1 , z ∈ C1. (1)The corresponding inequality for an interval is|P ′(x)| ≤ n√1− x2 ||P ||[−1,1], −1 < x < 1, (2)and for the uniform norm of the derivative we have the so called Markoff in-equality‖P ′n‖[−1,1] ≤ n2‖Pn‖[−1,1].In this paper we prove the following analogue for circular arcs. Let 0 < ω ≤ piand letKω = {eiθ θ ∈ [−ω, ω]} (3)be the circular arc on the unit circle of central angle 2ω and with midpoint at1.Theorem 1 If Pn is a polynomial of degree at most n, then|P ′n(eiθ)| ≤n2(1 +√2 cos(θ/2)√cos θ − cosω)‖Pn‖Kω , θ ∈ (−ω, ω). (4)∗AMS Classification 42A05, Keywords: polynomial inequalities, circular arc, Bernsteininequality†Supported by ERC grant No. 2670551This is sharp:Theorem 2 For every θ ∈ (−ω, ω) there are nonzero polynomials Pn of degreen = 1, 2, . . . such that|P ′n(eiθ)| ≥ (1− o(1))n2(1 +√2 cos(θ/2)√cos θ − cosω)‖Pn‖Kω . (5)Of course, the ω = pi case is just the original Bernstein inequality (1). Also, ifwe write up the consequence for an arc on the circle RC1−R = {z |z+R| = R}:|P ′n(R(1− eiθ))| ≤n2R(1 +√2 cos(θ/2)√cos θ − cosω)‖Pn‖RKω−R, θ ∈ (−ω, ω),(6)apply it with ω = 1/R and θ = x/R, x ∈ [−1, 1], and let R → ∞, then weobtain (2) (with a change of variable) since|P ′n(R(1− eiθ))| → |P ′n(ix)|, ‖Pn‖RKω−R → ‖Pn‖[−i,i]and2R2(cos(x/R)− cos(1/R))→ 1− x2.The inequality in Theorem 4 can be written in alternative forms using theequilibrium measure νKω of Kω and the Green’s function g(z) = gC\Kω (z,∞)with pole at infinity of the complement of Kω (see [10], [4] or [11] for theseconcepts). In fact, if dν(z)/ds is the density (Radon-Nikodym derivative) of theequilibrium measure νKω with respect to arc length on C1, then (4) is the sameas|P ′n(ζ)| ≤n2(1 + 2pidνKω (ζ)ds)‖Pn‖Kω , ζ ∈ Kω, (7)and if g′±(ζ) denote the normal derivatives of the Green’s function in the direc-tion of the two normals to Kω, then another equivalent form is|P ′n(ζ)| ≤ nmax{g′−(ζ), g′+(ζ)}‖Pn‖Kω , ζ ∈ Kω. (8)For (7) and (8) see the proof of Theorem 1. We believe that this last form (witha factor (1 + o(1))) should be the correct form of the Bernstein inequality onsmooth Jordan curves. Our proof for Theorem 2 shows that if Kω is replacedby any C2 Jordan curve or Jordan arc, or even by a family of these, then anestimate better than (8) cannot be given, i.e. that the asymptotic Bernsteinfactor is at least as large as nmax{g′−(ζ), g′+(ζ)}.We also mention the Markoff type inequality: if Pn is a polynomial of degreeat most n, then||P ′n||Kω ≤ (1 + o(1))n22cot(ω2)||Pn||Kω . (9)2This is sharp again: for some nonzero polynomials Pn we have||P ′n||Kω ≥ (1− o(1))n22cot(ω2)||Pn||Kω . (10)These are immediate consequences of [3], p. 243, see Section 2.For even n Theorem 1 is an easy consequence of the classical Videnskiiinequality on trigonometric polynomials, and for odd n it also follows froma related inequality of Videnskii for a trigonometric expression in which thefrequencies of cosine and sine are an integer plus one half. This derivation will bedone in the next section. The proof of Theorem 2 in section 5 will be based on atheorem of [6] for Bernstein-type inequalities on a Jordan curve (homeomorphicimage of the unit circle). In the process we shall need to calculate the normalderivatives of the Green’s function of the complement of C \Kω, which will bedone in section 3. Once this is done, we give in section 4 a relatively simpledirect proof for Theorem 1 using a result of Borwein and Erde´lyi.2 Theorem 1 and Videnskii’s inequalitiesLetV (θ) = V (ω; θ) =√2 cos(θ/2)√cos θ − cosω =cos(θ/2)√sin2(ω2)− sin2 ( θ2) . (11)The classical Bernstein inequality for trigonometric polynomials was ex-tended by Videnskii (see e.g. [3], Ch. 5, E.19, p. 242 or [14]): let Qm(t)be a trigonometric polynomial with real coefficients of degree at most m, andlet ω ∈ (0, pi). Then for any θ ∈ (−ω, ω), we have|Q′m(θ)| ≤ mV (ω; θ)||Q||[−ω,ω]. (12)There is an extension to half-integer trigonometric polynomials [15]: letQm+1/2(t) =m∑j=0aj cos((j +12)t)+ bj sin((j +12)t), aj , bj ∈ R.Then for any θ ∈ (−ω, ω), we have|Q′m+1/2(θ)| ≤(m+12)V (ω; θ)||Qm+1/2||[−ω,ω]. (13)Standard trick leads to the same inequalities with complex coefficients: forexample, if Q˜m is a trigonometric polynomial with complex coefficients andθ ∈ (−ω, ω), then let |τ | = 1 be such that τQ˜′m(θ) = |Q˜′m(θ)|. Now if we apply(12) to the real trigonometric polynomial Qm(t) = ℜ(τQ˜m(t)) then we get (12)for Q˜m.3Proof of Theorem 1. Let Pn be an algebraic polynomial of degree at mostn and setQn/2(t) := e−in2tPn(eit). (14)For this||Qn/2||[−ω,ω] = ||Pn||K ,andQ′n/2 (θ) = e−in2θ (−in/2)Pn(eiθ) + e−in2 θP ′n(eiθ)eiθi. (15)So|P ′n(eiθ)| ≤ |Q′n/2 (θ) |+n2|Pn(eiθ)|, θ ∈ (−ω, ω)and (4) is an immediate consequence of (12) (in the case when n is even) and(13) (when n is odd) with m = n/2, because the second term on the right is≤ ‖Pn‖Kω .Since (15) gives for t ∈ (−ω, ω)∣∣∣|Q′n/2(t)| − ∣∣P ′n (eit) e−in2 t∣∣∣∣∣ ≤ ||Pn||Kω n2 , (16)(9) follows from the following inequality of Videnskii (see e.g. [3], p. 243): ifQm(t) is a trigonometric polynomial of degreem, then for 2m ≥(3 tan2(ω2)+ 1)1/2,||Q′m||[−ω,ω] ≤ 2m2 cotω2||Qm||[−ω,ω]. (17)Indeed, we may assume that n is even (if it is odd, consider Pn as a polynomialof degree at most n+ 1), and then we can apply (17) to the Qn/2 in (14) (notethat now the term on the right of (16) is o(n2)).Since (17) is sharp (see [3], p. 243), (10) also follows.3 The normal derivatives of the Green’s func-tionLet K = Kω. Denote the Green’s function of C \ K with pole at infinity byg(ζ), g(ζ) = gC\K(ζ,∞). There are two normals to K, the “outer” normal ispointing into the exterior of the unit circle, and the “inner” normal is pointingtowards its interior. We need to compute the normal derivatives g+, g− of gwith respect to both normals.Denote the equilibrium measure of K by ν. It is known that ν is absolutelycontinuous with respect to arc length, see [11], p. 209, Theorem 2.1. We denotethe density by dν/ds. Recall also in the next proposition the definition of Vfrom (11).4Proposition 3 Let ζ0 = eiθ0 be an inner point of K and g+, g− the normalderivatives of the Green’s function in the direction of the outer and inner normal.Theng′+(ζ0) =12(1 +√2 cos θ0/2√cos θ0 − cosω)(18)andg′−(ζ0) =12(−1 +√2 cos θ0/2√cos θ0 − cosω). (19)Corollary 4 We haveg′+(ζ0) + g′−(ζ0) = 2pidν(ζ0)ds, (20)g′+(ζ0) + g′−(ζ0) = V (θ0) (21)andg′+(ζ0)− g′−(ζ0) = 1. (22)Proof. Fix ζ0 = eiθ0 ∈ K, where θ0 ∈ (−ω, ω).Let g˜ be the analytic conjugate of g with the normalization g˜(e−iω) =limζ→e−iω g˜(ζ) = 0, and let G(z) = g(z) + ig˜(z) be the complex Green’s func-tion. Then, using the properties of Green’s functions, it is easy to see thatΨ(z) = exp(G(z)) maps C \K conformally onto the exterior of the unit circle.SetR(z) = −(z − eiω)(z − e−iω)and S(z) = i(z − 1). We cut the plane along the arc K, and take the branch ofthe square root√R(z) which is positive at 0. Then√R(0) = 1 = iS(0).With these notations it was proved in [8, p. 398] thatG(z) =12∫ ze−iω1ζ(1− iS(ζ)√R(ζ))dζ,where the integration is along a path from e−iω to z that does not intersect K.Now if ζ0 = eiθ0 is an inner point of K, theng′+(ζ0) = ℜ∂G(ζ0)∂n+= ℜ ζ0G′(ζ0+) = ℜ 12(1− iS(ζ0+)√R(ζ0+)), (23)where ζ0+ indicates that the appropriate value is taken on the outer side of K(which is the side that lies outside the unit disk), whileg′−(ζ0) = ℜ∂G(ζ0)∂n−= −ℜ ζ0G′(ζ0−) = ℜ 12(iS(ζ0−)√R(ζ0−)− 1), (24)5and here ζ0− indicates that the appropriate value is taken on the inner side ofK. Here, for ζ0 = eiθ0 lying in the inner side of K, we haveR(eiθ0) = −(eiθ0 − eiω)(eiθ0 − e−iω) = −eiθ0 (eiθ0 − 2 cosω + e−iθ0)= −2eiθ0 (cos t0 − 2 cosω) ,and henceiS(ζ0−)√R(ζ0−)=1− eiθ0−eiθ0/2√2(cos θ0 − cosω)=√2 cos θ0/2√cos θ0 − cosωis real and positive. In a similar vein, for ζ0 = eiθ0 lying in the outer side of Kwe haveiS(ζ0+)√R(ζ0+)= − iS(ζ0−)√R(ζ0−)= −√2 cos θ0/2√cos θ0 − cosω.Plugging these into (23)–(24) we get (18) and (19).From these formulae (21) and (22) immediately follow. Formula (20) isknown, see e.g. [11], Theorem 2.3, p. 211.Corollary 5 Let Ψ be a conformal map from C \ K onto the exterior of theunit disk. Then for ζ0 = eiθ0 lying in the interior of K we haveg′+ (ζ0) =V (θ0) + 12= |Ψ′(ζ0+)|. (25)The derivative on the right-hand side is understood from the outside of ∆ (bythe Kellogg-Warschawski theorem Ψ′(ζ0+) exists on the boundary in the sensethat Ψ′(ζ) has a limit as ζ → ζ0 from the outside, see [9], Theorems 3.5, 3.6).Note also that different Ψ’s differ by a multiplicative constant of modulus 1,so it does not matter which one we take.Proof. The first equality has been verified in Proposition 3 and Corollary 4.In the proof of Proposition 4 we have also seen thatg′+(ζ0) = ℜ ζ0G′(ζ0+) = ℜΨ′(ζ0+)ζ0Ψ(ζ0+).Now at ζ0 the direction of the outer normal toK is ζ0, so (using the conformalityof Ψ) Ψ′(ζ0+)ζ0/|Ψ′(ζ0+)| is the direction of the outer normal to C1 at thepoint z = Ψ(ζ0+), but this direction is again the same as z = Ψ(ζ0+). As aconsequence, Ψ′(ζ0+)ζ0/Ψ(ζ0+) is positive, and hence we have the formulag′+(ζ0) = ℜΨ′(ζ0+)ζ0Ψ(ζ0+)=∣∣∣∣ℜΨ′(ζ0+)ζ0Ψ(ζ0+)∣∣∣∣ = |Ψ′(ζ0+)|. (26)64 A direct proof for Theorem 1In this section we prove Theorem 1 using the following result of P. Borwein andT. Erde´lyi (see [3], p. 324, Theorem 7.1.7). Recall that we denote the unit diskby ∆ and the unit circle by C1. Let ak ∈ C \ C1, k = 1, . . . ,m, setB+m(z) :=∑k:|ak|>1|ak|2 − 1|ak − z|2 , B−m(z) :=∑k:|ak|<11− |ak|2|ak − z|2 ,and letBm(z) := max(B+m(z), B−m(z)).Then, for every rational function r(z) of the form r(z) = Q(z)/∏mk=1(z − ak)where Q is a polynomial of degree at most m, we have|r′(z)| ≤ Bm(z)||f ||C1 z ∈ C1. (27)We shall need the functionζ = Φ(z) = z1 + z sin (ω/2)z + sin (ω/2). (28)Simple computation gives, as e.g. in [7] p. 369 equation (4), that Φ is a con-formal map from the complement of the unit disk onto C \K, so Ψ = Φ−1 isone of the Ψ’s in Corollary 5. It is also easy to see that if ℜz > − sin(ω/2),then ζ = Φ(z) lies on the outer side of the arc K (i.e. then ζ = ζ+), while ifℜz < − sin(ω/2), then ζ = Φ(z) lies in the inner side of the arc K (i.e. in thiscase ζ = ζ−).Without loss of generality, we may assume that the polynomial in Theorem1 is of the form Pn(ζ) = (ζ − α1) . . . (ζ − αn) (i.e. it has leading coefficient 1)and definer(z) := Pn(1Φ(z)), (29)where Φ is the function from (28). Then‖r‖C1 = ‖Pn‖Kand (see (28))r(z) =n∏j=1(z + sin (ω/2)z (1 + z sin(ω/2))− αj)=∏nj=1(−αj sin (ω2 ) z2 + (1− αj)z + sin (ω2 ))zn(z sin(ω2)+ 1)n .So, to use (27) we set m = 2n, a1 = . . . = an = 0, and an+1 = . . . = a2n =−1/ sin (ω2 ). For z = eit we see thatB−2n(z) = n and B+2n(z) = n∣∣∣ −1sin(ω/2) ∣∣∣2 − 1∣∣∣ −1sin(ω/2) − eit∣∣∣27and here the second term isB+2n(z) = ncos2 (ω/2)|1 + sin (ω/2) cos t+ i sin (ω/2) sin t|2= ncos2 (ω/2)1 + sin2 (ω/2) + 2 sin (ω/2) cos t.Taking maximum, we getB2n(z) ={n, if ℜ(z) = cos(t) ≥ − sin (ω/2) ,n cos2(ω/2)1+sin2(ω/2)+2 sin(ω/2) cos t, if ℜ(z) = cos(t) ≤ − sin (ω/2) ,(30)and it is important to note that B2n(z) = B−2n(z) = n (first line), if ζ = Φ(z) is”from the outer side” of K. Hence the Borwein-Erde´lyi inequality implies that∣∣∣∣P ′n(1Φ(z))Φ′(z)Φ2(z)∣∣∣∣ ≤ B2n(z)||Pn||K ,and since here |Φ(z)| = 1 for z ∈ C1, we get for Φ(z) = ζ =: eiθ, θ ∈ (−ω, ω)∣∣P ′n (e−iθ)∣∣ ≤ B2n(z)|Φ′(z)| ||Pn||K . (31)For each θ ∈ (−ω, ω), there are two z ∈ C1 such that Φ(z) = eiθ, one on thearc in the half-plane {z ℜz ≥ − sin(ω/2)}, and one on the complementary arcof C1. We choose the former one in (31), which corresponds to the first line in(30), and get ∣∣P ′n (e−iθ)∣∣ ≤ n|Φ′(z)| ||Pn||K . (32)Since Ψ (Φ (z)) = z, we have Ψ′ (Φ (z)) Φ′ (z) = 1, i.e. |Ψ′(ζ)| = 1/|Φ′(z)|. Ifwe substitute this into (32) and use Corollary 5, we obtain (4) (note also thatV (−θ) = V (θ)).5 Proof of Theorem 2It was proven in [6], Theorem 1.3, 1.4, that if Γ is a C2 smooth Jordan curve(homeomorphic image of the unit circle), Ω is the unbounded component of itscomplement and gΩ(z,∞) is the Green’s function in Ω with pole at infinity, then|P ′n(ζ)| ≤ (1 + o(1))n∂gΩ(ζ,∞)∂n‖Pn‖Γ, ζ ∈ Γ,where n is the inner normal to Γ with respect to Ω. Furthermore, this is sharp,for if ζ ∈ Γ is given, then there are nonzero polynomials Pn with|P ′n(ζ)| ≥ (1− o(1))n∂gΩ(ζ,∞)∂n‖Pn‖Γ. (33)8Now consider K and a point ζ on K which is not one of the endpoints of K.We augment K to a C2 smooth Jordan curve Γ by attaching a small domain(as the interior of Γ) to K that lies in the unit disk (see Figure 1).KzGFigure 1: The domain attached to KWe can do that in such a way that if ε > 0 is given, then∂gΩ(ζ,∞)∂n≥ (1− ε)∂gC\K(ζ,∞)∂n= (1− ε)g′+(ζ). (34)In fact, since K is part of Γ we have gΩ(ζ,∞) ≤ gC\K(ζ,∞), and at infinity thedifference gΩ(ζ,∞) − gC\K(ζ,∞) coincides with log(cap(Γ)/cap(K)) (see [10],Theorem 5.2.1), where cap(·) denotes logarithmic capacity. As we shrink Γ toK,the capacity of Γ tends to the capacity of K, and so the nonnegative harmonicfunction gΩ(ζ,∞)− gC\K(ζ,∞) tends to zero at infinity (this difference is alsoharmonic there). Now we get from Harnack’s theorem ([10], Theorems 1.3.1 and1.3.3) that this difference tends to 0 uniformly on compact subsets of C \ K,and then (34) will be true if Γ is sufficiently close to K by [6], Lemma 7.1.Now apply (33) to this Γ. For the corresponding polynomials Pn we canwrite, in view of ||Pn||K ≤ ||Pn||Γ,|P ′n(ζ)| ≥ (1− o(1))n∂gΩ(ζ,∞)∂n‖Pn‖Γ ≥ (1− o(1))n(1− ε)g′+(ζ)||Pn||K .Since here ε > 0 is arbitrary, and by Corollary 25 the last factor on the right-hand side is (1 + V (θ))/2 with ζ = eiθ, the proof is complete.9References[1] M. J. Ablowitz and A. S. Fokas, Complex variables: introduction and applica-tions, Second edition, Cambridge Texts in Applied Mathematics, CambridgeUniversity Press, Cambridge, 2003.[2] M. Baran, Bernstein type theorems for compact sets in Rn, J. Approx.Theory 69 (1992) 156–166.[3] P. Borwein and T. Erde´lyi, Polynomials and polynomial inequalities, Grad-uate Texts in Mathematics, 161, Springer Verlag, New York, 1995.[4] J. B. Garnett and D. E. Marshall, Harmonic measure, New MathematicalMonographs 2, Cambridge University Press, Cambridge, 2008.[5] C. K. Kobindarajah and D. S. Lubinsky, Lp Markov-Bernstein inequalitieson all arcs of the circle , J. Approx Theory 116 (2002) 343–368.[6] B. Nagy and V. Totik, Sharpening of Hilbert’s lemniscate theorem, J. Anal.Math. 96 (2005) 191–223.[7] A. V. Olesov, On the application of conformal mappings to inequalities fortrigonometric polynomials, Mat. Zametki 76 (2004) 396–408.[8] F. Peherstorfer and R. Steinbauer, Strong asymptotics of orthonormal poly-nomials with the aid of Green’s function, SIAM J. Math. Anal.. 32(2000),385–402.[9] Ch. Pommerenke, Boundary Behavior of Conformal Mappings, Grundlehrender mathematischen Wissenschaften, 299, Springer Verlag, Berlin, Heidel-berg New York, 1992.[10] T. Ransford, Potential Theory in the Complex plane, Cambridge UniversityPress, Cambridge, 1995.[11] E. B. Saff and V. Totik, Logarithmic potentials with external fields, Grund-lehren der mathematischen Wissenschaften, 316, Springer Verlag, Berlin,1997.[12] B. Simon, Orthogonal polynomials on the unit circle. Part 2, AmericanMathematical Society Colloquium Publications, 54, American MathematicalSociety, Providence, RI, 2005.[13] V. Totik, Polynomial inverse images and polynomial inequalities, ActaMath. 187 (2001) 139–160.[14] V. S. Videnskii, Extremal estimates for the derivative of a trigonometricpolynomial on an interval shorter than its period, Soviet Math. Dokl. 1(1960) 5–8.10[15] V. S. Videnskii, On trigonometric polynomials of half-integer order, Izv.Akad. Nauk Armjan. SSR Ser. Fiz.-Mat. Nauk 17 (1964) 133–140.Be´la Nagy,Bolyai InstituteAnalysis Research Group of the Hungarian Academy of SciencesUniversity of SzegedSzegedAradi v. tere 1, 6720, Hungarynbela@math.u-szeged.huVilmos TotikBolyai InstituteAnalysis Research Group of the Hungarian Academy of SciencesUniversity of SzegedSzegedAradi v. tere 1, 6720, HungaryandDepartment of MathematicsUniversity of South Florida4202 E. Fowler Ave, PHY 114Tampa, FL 33620-5700, USAtotik@math.usf.edu11

Bernstein's inequality for algebraic polynomials on circular arcs

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Bernstein's inequality for algebraic polynomials on circular arcs

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