A note on the bounds of the error of Gauss Turán-type quadratures

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1 Journal of Computational and Applied Mathematic A note on the bound of the error of Gau Turán-type quadrature Gradimir V. Milovanović a, Miodrag M. Spalević b, a Department of Mathematic, Faculty of Electronic Engineering, Univerity of Niš, P.O. Box 73, 8 Niš, Serbia and Montenegro b Department of Mathematic and Informatic, Faculty of Science, Univerity of Kragujevac, P.O. Box 6, 34 Kragujevac, Serbia and Montenegro Received 2 September 25; received in revied form 22 December 25 Abtract Thi note i concerned with etimate for the remainder term of the Gau Turán quadrature formula, R n, f = wtftdt n 2 A i, f i τ, = i= where wt = U n t/n 2 t 2 i the Gori Michelli weight function, with U n t denoting the n th degree Chebyhev polynomial of the econd kind, and f i a function analytic in the interior of and continuou on the boundary of an ellipe with foci at the point ± and um of emiaxe ϱ >. The preent paper generalize the reult in [G.V. Milovanović, M.M. Spalević, Bound of the error of Gau Turán-type quadrature, J. Comput. Appl. Math ], which i concerned with the ame problem when =. 26 Elevier B.V. All right reerved. MSC: Primary 65D3, 65D32; econdary 4A55 Keyword: Gau Turán quadrature formula; Gori Michelli weight function; Error bound for analytic function. Introduction Let w be an integrable weight function on the interval,. We conider the error term R n, f of the Gau Turán quadrature formula with multiple node wtftdt = n 2 A i, f i τ + R n, f, = i= The author were upported in part by the Swi National Science Foundation SCOPES Joint Reearch Project No. IB New Method for Quadrature and the Serbian Minitry of Science and Environmental Protection. Correponding author. Tel.: ; fax: addree: grade@elfak.ni.ac.yu G.V. Milovanović, pale@kg.ac.yu M.M. Spalević /$ - ee front matter 26 Elevier B.V. All right reerved. doi:.6/j.cam

2 G.V. Milovanović, M.M. Spalević / Journal of Computational and Applied Mathematic which i exact for all algebraic polynomial of degree at mot 2 + n, and whoe node are the zero of the correponding -orthogonal polynomial π n, t of degree n. For more detail on Gau Turán quadrature and -orthogonal polynomial ee the book [] and the urvey paper [4]. Let be a imple cloed curve in the complex plane urrounding the interval [, ] and D be it interior. If the integrand f i an analytic function in D and continuou on D, then we take a our tarting point the well-known expreion of the remainder term R n, f in the form of the contour integral R n, f = 2πi The kernel i given by K n, zf z dz.. ϱ n, z K n, z =, z / [, ],.2 [π n, z] 2+ where [π n, t] 2+ ϱ n, z = wt dt, n N,.3 z t and π n, t i the correponding -orthogonal polynomial with repect to the weight function wt on,. The integral repreentation. lead to a general error etimate, by uing Hölder inequality, i.e., R n, f = 2π K n, zf z dz 2π /r K n, z r dz /r fz r dz, R n, f 2π K n, r f r,.4 where r +,/r + /r =, and { f r := fz r dz /r, r<+, max fz, r =+. z The cae r =+ r = give R n, f l max 2π K n,z max fz,.5 z z where l i the length of the contour. On the other ide, for r = r =+, the etimate.4 reduce to R n, f K n, z dz max 2π fz,.6 z which i evidently tronger than the previou, becaue of inequality K n, z dz l max K n,z. z Alo, the cae r = r = 2 could be of certain interet. For getting the etimate.5 or.6 it i neceary to tudy the magnitude of K n, z on or the quantity L n, := K n, z dz, 2π repectively ee, e.g., [5,6].

3 278 G.V. Milovanović, M.M. Spalević / Journal of Computational and Applied Mathematic Error etimate.6 for Gau Turán quadrature with Gori Micchelli weight function, and when i taken to be a confocal ellipe, are conidered for the general cae N in Section 2. The particular cae = wa conidered in [7]. 2. Error etimate for Gau Turán quadrature with Gori Micchelli weight function for general N Let the contour be an ellipe with foci at the point ± and um of emi-axe ϱ >, E ϱ ={z C : z = 2 ϱeiθ + ϱ e iθ, θ 2π}. 2. In [7] we conidered the error etimate.6 for Gau Turán quadrature formula with = and for the Gori Michelli weight function wt = w n t = U n 2 t n 2 t 2, 2.2 where U n co θ = in nθ/ in θ i the Chebyhev polynomial of the econd kind. Here we conider the general cae with N. It i well-known that for the weight function 2.2 the Chebyhev polynomial T n t of the firt kind appear to be -orthogonal one cf. [2]. For z E ϱ, i.e., z = 2 ξ + ξ, ξ = ϱe iθ,wehaveπ n, z = T n z = 2 ξn + ξ n and, according to.3 and 2.2, ϱ n, z = n 2 T n t 2+ U 2 n t z t Since dz =2 /2 a 2 co 2θ dθ, where we put t 2 dt. 2.3 a j = a j ϱ = 2 ϱj + ϱ j, j N, ϱ >, 2.4 we have, according to.2, L n, E ϱ = 2π 2π ϱ n, z a 2 co 2θ /2 2 T n z 2+ dθ. 2.5 Now, from 2.3, by ubtituting t = co θ, we have, in view of T n co θ = co nθ and U n co θ = in nθ/ in θ, ϱ n, z = π [co nθ] 2+ [in nθ] 2 n 2 dθ. z co θ ϱ n, z = We tranform [co nθ] 2+ by uing a formula from [3, Eq..32.7], while [in nθ] 2 = co 2nθ/2. Therefore, co2 + 2knθ co 2nθ i.e., = π n n k= k= 2+ k [ 2 + π k z co θ co2 + 2knθ z co θ [ 2 + π co2 + 2knθ ϱ n, z = n dθ k k= z co θ π co knθ dθ π co2 2knθ 2 z co θ 2 z co θ dθ π ] co2 + 2knθ co 2nθ dθ dθ, z co θ ] dθ.

4 G.V. Milovanović, M.M. Spalević / Journal of Computational and Applied Mathematic Furthermore, uing [3, Eq ], one find π co mθ z co θ dθ = π m, z z 2 m N, z 2 and we obtain ϱ n, z = 2 2+ n 2 [ {[ 2 + 2π k ξ ξ ξ 2 kn+n 2π 2 ξ ξ k= ξ 2 kn+3n 2 + 2π k ξ ξ ξ 2 kn n + 2π ξ ξ ξ n k= ]} 2 + 2π 2 + 2π + ξ ξ ξn ξn, ξ ξ where we ued that z 2 = 2 ξ ξ and z z 2 = ξ. Finally, we obtain π ϱ n, z = 2 2+ n 2 ξn ξ n b α, 2.6 ξ ξ where we ued the notation 2 + b b =, α α n, ϱ, θ = ξn ξ n Uing 2.6 and ξ n 2 + k ξ 2 kn. T n z =a 2n + co 2nθ /2 / 2, ξ k ξ k = 2a 2k co 2kθ /2 k N, the quantity 2.5 reduce to L n, E ϱ = 2 +2 n 2 2π a 2n co 2nθ b α 2 k= a 2n + co 2nθ 2+ dθ, 2.7 where b α 2 = b 2 2b Re{α}+ α 2 b R, α C. It i not difficult to conclude that α 2 = α α = h 2 2nθ, where h 2 θ = 2a 2n co θ ϱ 2n+ W ϱ n, θ 2 and W ϱ, θ := 2+ = ϱ 2 e i /2θ ha been defined in [6, Eq. 4.2]. Let x = ϱ 4n. Recall that W ϱ n, θ 2 = l= A l co lθ cf. [6, Eq ], where A = x / x = ] and A l = 2 x l/2 l 2 + = 2 + x, + l l=,...,.

5 28 G.V. Milovanović, M.M. Spalević / Journal of Computational and Applied Mathematic Further, we have { Re{α}=Re /ξ 2n where h θ = = Therefore, 2.7 become 2 + = ξ 2 n 2 + ϱ 2 n co θ L n, E ϱ = 2 +2 n 2 2π } = h 2nθ, 2 + = ϱ 2 n co + θ. a 2n co 2nθb 2 2bh 2nθ + h 2 2nθ a 2n + co 2nθ 2+ dθ. The lat integrand depend in θ via co 2nlθ n N,l {,..., + }, N. It i a continuou function of the form g2nθ, where gθ gco θ, co 2θ,...,co + θ. Becaue of periodicity, it i eay to prove that 2π g2nθ dθ = 2 π gθ dθ. Therefore, L n,e ϱ reduce to L n, E ϱ = 2 + n 2 π Further, h θ can be written in the form h θ = x /2 = a 2n co θb 2 2bh θ + h 2 θ a 2n + co θ 2+ dθ [x /2 co θ x /2 co + θ], i.e., after expanding the um and putting in order, 2 + h θ = + 2 l 2 + x l/2 co lθ. + l + + l Now, 2.8 obtain the form L n, E ϱ = π hn, 2 + n 2 ϱ, θ dθ, 2.9 where h n, ϱ, θ = β/a 2n + co θ 2+ and β β n, ϱ, θ = a 2n co θ 2x +/2 a 2n co θ A l co lθ l= l l + + l On the other hand, applying Cauchy inequality to 2.9, we obtain π π /2 L n, E ϱ 2 + n 2 h n, ϱ, θ dθ. x l/2 co lθ.

6 G.V. Milovanović, M.M. Spalević / Journal of Computational and Applied Mathematic Since + β = a 2n b 2 l ba 2n x l/2 co lθ + l + + l + + b 2 l 2 + co θ 4b co θ x l/2 co lθ + l + + l + 2x +/2 a2n 2 2a 2n co θ + co 2 θ A l co lθ, we have that π π β h n, ϱ, θ dθ = dθ 2+ a 2n + co θ + = a 2n b 2 l 2 + J + 4ba 2n x l/2 J l + + l + l + + b 2 J 2b l= l + l x l/2 J l + J l+ + l + x +/2 A l [2a2n 2 J l 2a 2n J l + J l+ + J l + ] 2 J l 2 + J l+2, l= where by J l we denoted the following integral cf. [6, p. 27]: π co lθ J l J l a 2n = dθ. 2+ a 2n + co θ It i well-known that ee [3, Eq ] or [6, Eq. 4.6] J l J l a 2n = 22+ π l x l/2 x = 2 + l x 2. l + Therefore, we have πγ L n, E ϱ 2 + n 2, 2. where γ γ n, ϱ = J x + 2 x J l x l/2 J l x + J l + J l+ + l + + l x [ + x +/2 x + 2 A l + J l x + J l + J l+ + ] J l 2 + J l+2. 2x x 2 l= In thi way, we have jut proved the following reult. Theorem 2.. Let E ϱ ϱ > be given by 2., a 2n be defined by 2.4, and x = ϱ 4n. Then, for the weight function 2.2, the quantity L n, E ϱ can be expreed in form 2.9. Furthermore, etimate 2. hold.

7 282 G.V. Milovanović, M.M. Spalević / Journal of Computational and Applied Mathematic a b Fig.. Log of the value L n, E ϱ olid line, with n = 5, given by 2.9 and it bound given by 2. dahed line for = the cae a and = 2 the cae b. Example 2.2. The function ϱ log L n, E ϱ, a well a it bound which appear on the right ide in 2., are given in Fig.. Bound 2. are very precie epecially for larger value of n,, ϱ. Acknowledgment We are thankful to the referee for a careful reading of the manucript and for their valuable comment. Reference [] A. Ghizzetti, A. Oicini, Quadrature Formulae, Akademie Verlag, Berlin, 97. [2] L. Gori, C.A. Micchelli, On weight function which admit explicit Gau Turán quadrature formula, Math. Comp [3] I.S. Gradhteyn, I.M. Ryzhik, Table of Integral, Serie, and Product, Academic Pre, New York, 98. [4] G.V. Milovanović, Quadrature with multiple node, power orthogonality, and moment-preerving pline approximation, in: W. Gautchi, F. Marcellan, L. Reichel Ed., Numerical Analyi 2, vol. V, Quadrature and orthogonal polynomial, J. Comput. Appl. Math [5] G.V. Milovanović, M.M. Spalević, Error bound for Gau Turán quadrature formulae of analytic function, Math. Comp [6] G.V. Milovanović, M.M. Spalević, An error expanion for Gau Turán quadrature and L -etimate of the remainder term, BIT [7] G.V. Milovanović, M.M. Spalević, Bound of the error of Gau Turán-type quadrature, J. Comput. Appl. Math