MONOTONICITY OF THE ERROR TERM IN GAUSS TURÁN QUADRATURES FOR ANALYTIC FUNCTIONS

Size: px
Start display at page:

Download "MONOTONICITY OF THE ERROR TERM IN GAUSS TURÁN QUADRATURES FOR ANALYTIC FUNCTIONS"

Transcription

1 ANZIAM J. 48(27), MONOTONICITY OF THE ERROR TERM IN GAUSS TURÁN QUADRATURES FOR ANALYTIC FUNCTIONS GRADIMIR V. MILOVANOVIĆ 1 and MIODRAG M. SPALEVIĆ 2 (Received March 19, 26) Abstract For Gauss Turán quadrature formulae with an even weight function on the interval ; 1] and functions analytic in regions of the complex plane which contain in their interiors a circle of radius greater than 1, the error term is investigated. In some particular cases we prove that the error decreases monotonically to zero. Also, for certain more general cases, we illustrate how to check numerically if this property holds. Some `2-error estimates are considered. 2 Mathematics subect classification: primary 41A55; secondary 65D3, 65D32. Keywords and phrases: Gauss Turán quadrature; error term; monotonicity; weight function; analytic functions. 1. Introduction Let w be an integrable weight function on the interval.; 1/. It is well known that the Gauss Turán quadrature formula with multiple nodes, w.t/ f.t/ dt n 2s A i; f.i/. / + R n;s. f /; (1.1) 1 i is exact for all algebraic polynomials of degree at most 2.s + 1/n 1 and that its nodes are the zeros of the corresponding s-orthogonal polynomial ³ n;s.t/ of degree n. For more details on Gauss Turán quadratures and s-orthogonal polynomials see the book 5] and the survey paper 9]. Numerically stable procedures for calculating the Gauss Turán quadrature formula have been proposed in 3, 1, 16]. 1 Department of Mathematics, University of Niš, Faculty of Electronic Engineering, P. O. Box 73, 18 Niš, Serbia; grade@ni.ac.yu. 2 Department of Mathematics and Informatics, University of Kraguevac, Faculty of Science, P. O. Box 6, 34 Kraguevac, Serbia; spale@kg.ac.yu. c Australian Mathematical Society 27, Serial-fee code /7 567

2 568 Gradimir V. Milovanović and Miodrag M. Spalević 2] Let be a simple closed curve in the complex plane surrounding the interval ; 1] and D be its interior. If the integrand f is an analytic function in D and continuous on D, then we take as our starting point the well-known expression of the remainder term R n;s. f / in the form of the contour integral R n;s. f / 1 K n;s.z/ f.z/ dz: (1.2) 2³i The kernel is given by K n;s.z/ % n;s.z/ ³n;s.z/ ] ; z ; 1]; 2s+1 where % n;s.z/ ³n;s.z/ ] 2s+1 w.t/ dt; n N; z t and ³ n;s.t/ is an s-orthogonal polynomial with respect to the measure w.t/ dt on.; 1/. Let w be one of the four generalized Chebyshev weight functions: (a) w 1.t/.1 t 2 / 2 ; (b) w 2.t/.1 t 2 / 12+s ; (c) w 3.t/.1 t/ t/ 12+s ; (d) w 4.t/.1 t/ 12+s.1 + t/ 2 : It is well known that the Chebyshev polynomials of the first kind T n are s-orthogonal subect to w 1.t/ on ; 1] for each s (see1]), and that for three other weights w i.t/, i 2; 3; 4, the s-orthogonal polynomials can be identified as Chebyshev polynomials of the second, third and fourth kinds: U n, V n and W n, which are defined by (see, for example, 18]) U n.t/ sin.n + 1/ cos.n + 12/ ; V n.t/ sin cos 2 and W n.t/ sin.n + 12/ sin 2 ; respectively, where t cos. However, such weights depend on s. The weight function in (d) can be omitted from investigation because W n. t/./ n V n.t/. Gori and Micchelli 6] have introduced for each n a class of weight functions defined on ; 1] for which the explicit Gauss Turán quadrature formulae can be found for all s. In other words, these classes of weight functions have the peculiarity that the corresponding s-orthogonal polynomials, of the same degree, are independent of s. This class includes certain generalized Jacobi weight functions w n;¼.t/ U n.t/ n 2¼+1.1 t 2 / ¼ ;

3 3] Monotonicity of the error term in Gauss Turán quadratures for analytic functions 569 where U n.cos / sin nsin (a Chebyshev polynomial of the second kind) and ¼>. In this case, the Chebyshev polynomials T n.t/ appear to be s-orthogonal polynomials. Using the representation (1.2) of the remainder term R n;s. f /, the results from 4, 8, 19] for Gauss quadrature formulae, and taking to be a confocal ellipse or circle, some very precise error bounds of R n;s. f / are derived in 11, 12, 13, 14]. The case with strong singularities has been considered in 15]. The monotonicity of quadrature approximations for some quadrature rules has been considered by several authors. For example, Newman 17] showed that, for a certain class of functions, the convergence of the general Newton-Cotes quadrature is monotone. Also, under the condition f.2n/, Brass 2] considered the quadrature rule Q n. f /, with n nodes, and proved certain monotonicity results in the case of Gaussian and Newton-Cotes rules. In this paper we consider the error term R n;s. f / in the Gauss Turán quadrature formula, with an even weight function on the interval ; 1], for functions analytic in regions of the complex plane which contain in their interiors a circle of radius greater than 1. For some important particular cases we prove that R n;s. f / decreases monotonically to zero. Also, we investigate numerically some more general cases and analyze some `2-error estimates. The paper is organized as follows. In Section 2 we give some preliminary results. The monotonicity of the error term is given in Section 3, including a numerical analysis. Finally, the `2-error estimates are considered in Section Preliminaries Let f be an analytic function in {z : z < 1 + 2"},where">, that is, f.z/ k a k z k ; z < 1 + 2": (2.1) Let ³ n;s.t/ be the s-orthogonal polynomial of the nth degree subect to the given weight function w.t/: ³ n;s.t/ n t n + n t n +,wherewetake n n.s/ >. In the following, we suppose that w.t/ is even, that is, w. t/ w.t/ on ; 1], and that the contour is the circle of radius 1 + ". The results which will be presented in this paper are a continuation of the results from 13]. Because of that we need some facts from 13], and we give them in this section again (without proofs).

4 57 Gradimir V. Milovanović and Miodrag M. Spalević 4] LEMMA 2.1. In the expansion + 1 ³ n;s.z/ b.s/ n; z n ; z 1; for n > 1 it holds that b.s/ n;2 +1, b.s/ n;2 >. ; 1; 2;:::/, and for n 1 it holds that 1³ 1;s.z/ b 1;z.s/, where b.s/ 1; 1 1 >. Therefore, in the case with even weight function, we have + 1 ³ n;s.z/ b.s/ n;2 z n 2 ; z 1; and ( 1 ³n;s.z/ ] + 2s+1 z n.2s+1/ ) 2s+1 b.s/ n;2 z 2 b.s/ n;2 z n.2s+1/ : It is clear that b.s/.s/ n;2 +1 andb n;2 >. If b.s/ n;2k are known, by using 7, Equation.314], it is possible to determine b.s/ n;2 in the following way: b.s/ n; b2s+1 n; ; b.s/ n;2 1 b.s/ n; LEMMA 2.2. For % n;s.z/ it holds that % n;s.z/ k1.2k.s + 1/ / b.s/ n;2k b.s/ n;2 2k : c.s/ n;2 z n 2 ; z > 1; where c.s/ n;2 >,n ; 1; 2;:::.s N /; ; 1; 2;:::. THEOREM 2.3. Let R n;s. f / be the remainder term in the Gauss Turán quadrature formula (1.1) and n;k k b.s/ n;2 c.s/ n;2k 2. Then, for n 1, R n;s. f / k a 2n.s+1/+2k n;k : (2.2) Let ¼ s be the moments of the weight function w.t/, thatis,¼ w.t/t dt ( ; 1; 2;:::). At the same time we introduce the quantities ¼ n 2s 1 i A i; ( i )! i ; ; 1; 2;::: ; which are obtained by applying the Gauss Turán quadrature sum on t, and which can be numerically calculated by using the above mentioned procedures. Because n;k >, we also obtained the following consequences of Theorem 2.3.

5 5] Monotonicity of the error term in Gauss Turán quadratures for analytic functions 571 COROLLARY 2.4. We have n;k ¼ 2n.s+1/+2k ¼ 2n.s+1/+2k ; k ; 1; 2;::: ; and n;k ¼ 2n.s+1/+2k 1 + o.1/] : Therefore, there exists k such that for all k k it holds that ¼ 2n.s+1/+2k >. Also, lim k + n;k, and the sequence ( ) n;k is bounded. k;1;2;::: 3. Monotonicity of the error term In this section we prove that the remainder R n;s. f / decreases monotonically to zero in some particular cases, when all a 2k in the expansion (2.1) are nonnegative. We use an analogous way of concluding as in Stenger 19, Lemma 4, Theorem 2] for the Gaussian quadrature formula (s ). 1. The case when n is fixed. We have R n;s. f / R n;s+1. f / k a 2n.s+1/+2k n;k k a 2n.s+2/+2k e.s+1/ n;k a 2n.s+1/ n; + +a 2n.s+1/+2n 2 n;n + a 2n.s+1/ n; + +a 2n.s+1/+2n 2 n;n + kn k a 2n.s+1/+2k n;k a 2n.s+2/+2k e.s+1/ n;k k ( ) a 2n.s+1/+2n+2k en;k+n.s/ e.s+1/ n;k : It will hold that R n;s. f / R n;s+1. f /, s ; 1; 2;:::, if it holds that n;k+n e.s+1/ n;k ; k ; 1; 2;::: : (3.1) Let us show that (3.1) holds if n 1; 2; 3, and w.t/ belongs to the class of Gori Micchelli weight functions. We have n;k+n e.s+1/ n;k 1 w.t/ z 2n.s+1/+2n+2k ³n;s.t/ ] 2s+1 2³i z t ³n;s.z/ ] dtdz 2s+1 1 w.t/ z 2n.s+2/+2k ³n;s+1.t/ ] 2s+3 2³i z t ³n;s+1.z/ ] dtdz : 2s+3

6 572 Gradimir V. Milovanović and Miodrag M. Spalević 6] After a little reordering we obtain n;k+n e.s+1/ n;k 1 2³i z 2n.s+2/+2k Tn.z/ ] 2s+3 w.t/ T n.t/ ] 2s+1 Sn.z; t/.t n.z/ + T n.t// ] dtdz; where we put S n.z; t/.t n.z/ T n.t//.z t/. Finally, because of orthogonality, n;k+n e.s+1/ n;k 1 z 2n.s+2/+2k 1 2³i Tn.z/ ] w.t/ T 2s+3 n.t/ ] 2s+2 S n.z; t/ dtdz : The value of the last expression will be equal to the coefficient of z when the expression under the integral over is expanded in the series of z. It is clear that all the coefficients of the series, which have been obtained by expanding z 2n.s+2/+2k T n.z/] 2s+3, are positive. Since the weight function is even, the corresponding (s-) orthogonal polynomial of degree n has the form (here this is the Chebyshev polynomial of the first kind) T n.t/ Þ n t n Þ n 2 t n 2 + Þ n 4 t n 4 ; where n Þ n > ; Þ n 2 > ; Þ n 4 > ; :::. Let n 1. Then T 1.t/ Þ 1 t and S.z; t/ Þ 1 >. Therefore w.t/ T 1.t/ ] 2s+2 S.z; t/ dt Þ 1 w.t/ T 1.t/ ] 2s+2 dt > : Let n 2. Then T 2.t/ Þ 2 t 2 Þ and S 1.z; t/ Þ 2.z + t/. Therefore w.t/ T 2.t/ ] 2s+2 S 1.z; t/ dt Þ 2 z w.t/ T 2.t/ ] 2s+2 dt ; which gives that the coefficient of z is positive. Let n 3. Then T 3.t/ Þ 3 t 3 Þ 1 t and S 2.z; t/ T 3.z/z + Þ 3 t 2 + Þ 3 tz. Therefore T 3.z/ z Now, because of w.t/ T 3.t/ ] 2s+2 S 2.z; t/ dt w.t/ T 3.t/ ] 2s+2 dt + Þ 3 w.t/t 2 T 3.t/ ] 2s+2 dt : w.t/ T 3.t/ ] 2s+2 dt > and Þ 3 w.t/t 2 T 3.t/ ] 2s+2 dt > ;

7 7] Monotonicity of the error term in Gauss Turán quadratures for analytic functions 573 we conclude that the coefficient of z is positive. It is not possible, in this way, to prove positivity of the coefficient of z for n The case when s is fixed. Consider the generalized Chebyshev weight functions whichareeven,thatis,w 1.t/ and w 2.t/. We have R n;s. f / R n+1;s. f / k a 2n.s+1/+2k n;k k a 2.n+1/.s+1/+2k n+1;k a 2n.s+1/ n; + +a 2n.s+1/+2s n;s + ks+1 a 2n.s+1/+2k n;k k a 2n.s+1/ n; + +a 2n.s+1/+2s n;s + k a 2.n+1/.s+1/+2k n+1;k a 2n.s+1/+2.s+1/+2k ( e.s/ n;k+s+1 n+1;k) : It will hold that R n;s. f / R n+1;s. f /, n ; 1; 2;:::, if it holds that We have n;k+s+1 n+1;k 1 2³i 1 2³i ( 2s i n;k+s+1 n+1;k ; k ; 1; 2;::: : (3.2) w.t/ z 2n.s+1/+2.k+s+1/ z t ( ) 2s+1 ³n;s.t/ ³ n;s.z/ ( ) 2s+1 ] ³n+1;s.t/ dtdz ³ n+1;s.z/ w.t/ z 2n.s+1/+2.k+s+1/ ³ n;s.t/³ n+1;s.z/ ³ n+1;s.t/³ n;s.z/ ³ n;s.z/] 2s+1 ³ n+1;s.z/] 2s+1 z t ) ³ n;s.t/³ n+1;s.z/] 2s i ³ n+1;s.t/³ n;s.z/] i Finally, by using the Christoffel-Darboux identity, n;k+s+1 n+1;k 1 2³i n+1 n z 2n.s+1/+2.k+s+1/ ³ n.z/] 2s+1 ³ n+1.z/] 2s+1 dtdz: n 2s ³ k.z/³ n+1.z/] 2s i ³ n.z/] i k w.t/³ k.t/³ n.t/] 2s i ³ n+1.t/] i dtdz; where ³ k is the corresponding ordinary orthogonal polynomial. i

8 574 Gradimir V. Milovanović and Miodrag M. Spalević 8] Consider now the case w.t/ w 1.t/, t.; 1/. By substituting t cos in the last integral we obtain n;k+s+1 n+1;k 1 2³i n+1 n z 2n.s+1/+2.k+s+1/ ³ n.z/] 2s+1 ³ n+1.z/] 2s+1 n 2s ³ k.z/³ n+1.z/] 2s i ³ n.z/] i k i cos kcos n] 2s i cos.n + 1/] i ddz: We see that n+1 n >, and that the expansions of z 2n.s+1/+2.k+s+1/ ; i ; 1;:::;2s; ³ n.z/] 2s+1 i ³ n+1.z/] i+1 in the series of z are such that all their coefficients are positive. Further, we use the following known formulae cos n] 2s i 2s i and cos.n + 1/] i i cos r.n + 1/, and put where a.i/ Both a.i/ Now, a.i/ r b.i/ r a.i/ cos2 n d and b.i/ r cos n] 2s i cos n d and b.i/ r b.i/ r cos2 r.n + 1/ d ; and b.i/ r can be found by using 7, Equation ]. cos kcos n] 2s i cos.n + 1/] i d s i i r 2s i 2s i a.i/ i r i r b.i/ r cos k cos n cos r.n + 1/ d a.i/ a.i/ b.i/ r a.i/ cos n cos.n + 1/] i cos r.n + 1/ d : cos.k + n/ + cos.k n/] cos r.n + 1/ d b.i/ r cos.k + n + r + rn/ + cos.k + n r rn/ + cos.k n + r + rn/ + cos.k n r rn/ ] d: (3.3) Since w.t/ w 1.t/ is an even weight function, the previous integrals are different from zero in those cases when k, i are both of even parity. Further, a.i/ ifi + is

9 9] Monotonicity of the error term in Gauss Turán quadratures for analytic functions 575 odd and b.i/ r ifi + r is odd. Therefore, we consider the cases in which k; i; ; r are all of even parity. The last integrals are of the form cos l d, l Z, and are equal to zero, except when l. Consider the case s 1. It is not difficult to conclude that the integral in (3.3) is not zero only in the cases when.k; i; ; r/ take the following values:.; ; ; /,.; 2; ; / and.1; 1; 1; 1/. In these cases we have: (i) If.k; i; ; r/.; ; ; /, then a./ cos 2 n d >; b./ d > : (ii) If.k; i; ; r/.; 2; ; /, then a.2/ d >; b./ cos 2.n + 1/ d > : (iii) If.k; i; ; r/.1; 1; 1; 1/, then a.1/ 1 cos 2 n d >; b.1/ 1 cos 2.n + 1/ d > : Therefore, we conclude that the requested coefficient of z will be positive. We derived the same conclusion in the case when w.t/ w 2.t/. Concluding in an analogous way we cannot see what happens if s 2. All that we can conclude is that R 1;2. f / R 2;2. f / Numerical analysis The formulae (3.1) and (3.2) give us a way to check numerically if monotonicity holds, for the error in the Gauss Turán quadrature formulae in the cases under consideration in this paper. Let us consider the Legendre weight function w.t/ w L.t/ 1on; 1]. Itis possible to display the graphs of n;k+n and e.s+1/ n;k for k M, form sufficiently large, by using programme packages such as MATLAB or MATHEMATICA, andso checking the conditions (3.1). The same can be said for the conditions (3.2). Let n be fixed, for instance, n 4. In Figure 1 we see the graphs of 4;k+4 (connected by dashed lines) and e.s+1/ 4;k (connected by solid lines), for s 1(left) and s 2 (right), where k ; 1; 2;:::;7. Finally let s be fixed, for instance, s 1. In Figure 2 we see the graphs of e.1/ n;k+2 (connected by dashed lines) and e.1/ n+1;k (connected by solid lines), for n 3(left)and n 4 (right), where k ; 1; 2;:::;6.

10 576 Gradimir V. Milovanović and Miodrag M. Spalević 1] FIGURE 1. The graphs show: R 4;1 R 4;2 (left) and R 4;2 R 4;3 (right) FIGURE 2. The graphs show: R 3;1 R 4;1 (left) and R 4;1 R 5;1 (right). 4. Some l 2 -error estimates For the remainder term in the Gauss Turán quadrature formula (1.1) we can derive the following estimate: Rn;s. f / Rn;s f : (4.1) If a given sequence ( ) n;k (n, s fixed) belongs to the space `p.p 1/, we k;1;::: apply Hölder s inequality to (2.2) to obtain Rn;s. f / ( + ) 1p ( ( ) + 1q e.s/ p n;k a2n.s+1/+2k q) ; (4.2) k where 1p + 1q 1. The quantities n;p.s/ ( + ( k e.s/ p ) 1p n;k) are independent of f and can be computed. The case when p + wasconsidered in 13]. k

11 11] Monotonicity of the error term in Gauss Turán quadratures for analytic functions 577 TABLE 1. Error constants W 12 n;s.w i/.i L; 2/ for some n; s and for the weight functions w L.t/ 1, w 2.t/ 1 t 2, t ; 1]. n W 12 n;1.w L / W 12 n;2.w L / W 12 n;1.w 2 / W 12 n;2.w 2 / 5 1:58./ 1:24./ 1:86. 2/ 1:11. 2/ 6 1:33./ 1:4./ 1:33. 2/ 7:88. 3/ 7 1:14./ 8:93. 2/ 9:99. 3/ 5:88. 3/ 8 1:1./ 7:84. 2/ 7:77. 3/ 4:55. 3/ 9 8:97. 2/ 6:98. 2/ 6:22. 3/ 3:63. 3/ 1 8:9. 2/ 6:3. 2/ 5:9. 3/ 2:96. 3/ Here we consider the case p q 2. The estimate (4.2) of type (4.1)gives R n;s ( + k ) 12 ( ) e.s/ 2 n;k W 12 : (4.3) n;s On the basis of the results from Section 2, there exists a k such that that is, < ¼ 2n.s+1/+2k ¼ 2n.s+1/+2k ; k k ; k + 1;:::; ¼ 2 2n.s+1/+2k ¼ 2 2n.s+1/+2k ; k k : Therefore, the series + ¼ 2 converges if the series + k 2n.s+1/+2k k ¼2 2k converges. But the last series converges if and only if the integral converges, since k I ¼ 2 2k + k w.t/w.y/ 1 ty w.t/t 2k dt dtdy w.y/y 2k dy I : Therefore, if the integral I converges, then the quantities Wn;s 12 Wn;s 12.w/ exist and can be calculated. The error constants Wn;s 12.w i/ (i L; 2) for some values of n, s and for weight functions w L.t/ 1, w 2.t/ 1 t 2, t ; 1], are displayed in Table 1. Numbers in parentheses indicate decimal exponents. These do not exist for the weight function w 1.t/ 1 1 t 2 (see, for example 19]). A calculation of W 12 n; in (4.3) for the Gaussian rule was given by Wilf 2] and Stenger 19, tableon page 157]. We finish, by deriving another expression for W n;s. Formula (4.2), for p q 2,

12 578 Gradimir V. Milovanović and Miodrag M. Spalević 12] has been derived as follows: R n;s. f / n 2s w.t/ f.t/ dt A i; f.i/. / 1 i ( 1 + ) ( 2s n + ).i/ w.t/ a t dt A i; a t 1 i ] 2s n a ¼ A i;.t /.i/ 1 i a.¼ ¼ / ( + ) 12 ( + ) 12 a 2.¼ ¼ / 2 : (4.4) Equality can be obtained in (4.4) by putting { ; ; 1;:::;2n.s + 1/ 1; a ¼ ¼ ; 2n.s + 1/; 2n.s + 1/ + 1;:::; (4.5) that is, for the function f.t/.¼ ¼ /t : (4.6) We have that (subect to (4.5), a always when is odd, since ¼ ¼ ) for the function f in (4.6) it holds that a a 2n.s+1/+2k n;k > ; 2n.s + 1/; 2n.s + 1/ + 2;::: : On the basis of these facts and (4.5) we conclude that a ; ; 1;::: : (4.7) By using (4.4), where a are given by (4.5), we have ] 1 2s n W n;s w.t/ ¼ A i;. i/! i t dt 1 i ( 2s n + ] ) 2s.i/ n A i; ¼ A i1; 1. i 1 /! i1 1 t : 1 i 11 i 1

13 13] Monotonicity of the error term in Gauss Turán quadratures for analytic functions 579 In fact, on the basis of (4.4) and (4.7), W n;s is equal to the last expression inside the absolute value, that is, W n;s w.t/ ( n 2s A i; 1 i ¼ + n 2s 1 i ¼ ] A i;. i/! i t dt n 2s 11 i 1 Let us put the last formula in order. First, we have w.t/ ¼ t dt w.t/ Second, we have w.t/t 2n.s+1/ dt ] ).i/ A i1; 1. i 1 /! i1 1 t : w.y/ ( ] t 2n.s+1/ y 2n.s+1/ + t 2n.s+1/+1 y + ) 2n.s+1/+1 dy dt w.t/w.y/.ty/ 2n.s+1/ dtdy : 1 ty w.t/ n 2s 1 i 2 A i; w.y/y 2n.s+1/.1 + ty + / dy n 2s 1 i n 2s 1 i A i; ] A i;. i/! i t dt ¼. i/! i + We now determine the last sum in this expression. Since.1 ¾/.l+1/ + ) l ¾ l, ¾ < 1, we have i! ( l. i/! i ¼ m2n.s+1/ i!. i/! i ¼. i/! i : w.y/y m dy

14 58 Gradimir V. Milovanović and Miodrag M. Spalević 14] 1 + ( ) 2n.s+1/ ( ] m i! w.y/y i. y/ i ). y/ m i dy i i Therefore 2 i! 2s n 1 i 2 Finally, we have 2s n 1 A i; i w.y/y i 2n.s+1/ dy.1 y/ i+1 + n 2s i! A i; 1 i i 11 i 1 i ¼. i/! i ( 2s n A i; A i1; 1 n 2s 1 ¼ 2 : i i ( ) ] i ¼ : i w.y/y i 2n.s+1/ dy.1 y/ i+1 + A i;. i/! i i ( ) ] i ¼ : i ). / 2. i/!. i 1 /! i i1 1 On the basis of the previous analysis, the quantity W n;s can be given in the form ] 2 where W n;s 2 W n;s W n;s + w.t/w.y/.ty/ 2n.s+1/ dtdy 1 ty 2s n i! A i; 1 i ¼ 2 ; w.y/y i 2n.s+1/ dy.1 y/ i+1 i ( ) ] i ¼ : i Acknowledgements The work was supported in part by the Serbian Ministry of Science and Environmental Protection.

15 15] Monotonicity of the error term in Gauss Turán quadratures for analytic functions 581 References 1] S. Bernstein, Sur les polynomes orthogonaux relatifs àunsegmentfini,j. Math. Pures Appl. 9 (193) ] H. Brass, Monotonie bei den Quadraturverfahren von Gauss and Newton Cotes, Numer. Math. 3 (1978) ] W. Gautschi and G. V. Milovanović, S-orthogonality and construction of Gauss Turán type quadrature formulae, J. Comput. Appl. Math. 86 (1997) ] W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 2 (1983) ] A. Ghizzetti and A. Ossicini, Quadrature Formulae (Akademie Verlag, Berlin, 197). 6] L. Gori and C. A. Micchelli, On weight functions which admit explicit Gauss Turán quadrature formulas, Math. Comp. 69 (1996) ] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 198). 8] D. B. Hunter, Some error expansions for Gaussian quadrature, BIT 35 (1995) ] G. V. Milovanović, Quadratures with multiple nodes, power orthogonality, and momentpreserving spline approximation, J. Comput. Appl. Math. 127 (21) ] G. V. Milovanović and M. M. Spalević, Quadrature formulae connected to -orthogonal polynomial, J. Comput. Appl. Math. 14 (22) ] G. V. Milovanović and M. M. Spalević, Error bounds for Gauss Turán quadrature formulae of analytic functions, Math. Comp. 72 (23) ] G. V. Milovanović and M. M. Spalević, Error analysis in some Gauss Turán Radau and Gauss Turán Lobatto quadratures for analytic functions, J. Comput. Appl. Math (24) ] G. V. Milovanović and M. M. Spalević, Bounds of the error of Gauss Turán-type quadratures, J. Comput. Appl. Math. 178 (25) ] G. V. Milovanović and M. M. Spalević, An error expansion for Gauss Turán quadratures and L 1 -estimates of the remainder term, BIT 45 (25) ] G. V. Milovanović and M. M. Spalević, Quadrature rules with multiple nodes for evaluating integrals with strong singularities, J. Comput. Appl. Math. 189 (26) ] G. V. Milovanović, M. M. Spalević and A. S. Cvetković, Calculation of Gaussian type quadratures with multiple nodes, Math. Comput. Modelling 39 (24) ] D. J. Newman, Monotonicity of quadrature approximations, Proc. Amer. Math. Soc. 42 (1974) ] A. Ossicini and F. Rosati, Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli, Boll. Unione Mat. Ital. 11 (1975) ] F. Stenger, Bounds on the error of Gauss-type quadratures, Numer. Math. 8 (1966) ] H. Wilf, Exactness conditions in numerical quadrature, Numer. Math. 6 (1964)

ERROR BOUNDS FOR GAUSS-TURÁN QUADRATURE FORMULAE OF ANALYTIC FUNCTIONS

ERROR BOUNDS FOR GAUSS-TURÁN QUADRATURE FORMULAE OF ANALYTIC FUNCTIONS MATHEMATICS OF COMPUTATION Volume, Number, Pages S 5-57(XX)- ERROR BOUNDS FOR GAUSS-TURÁN QUADRATURE FORMULAE OF ANALYTIC FUNCTIONS GRADIMIR V. MILOVANOVIĆ AND MIODRAG M. SPALEVIĆ This paper is dedicated

More information

On the remainder term of Gauss Radau quadratures for analytic functions

On the remainder term of Gauss Radau quadratures for analytic functions Journal of Computational and Applied Mathematics 218 2008) 281 289 www.elsevier.com/locate/cam On the remainder term of Gauss Radau quadratures for analytic functions Gradimir V. Milovanović a,1, Miodrag

More information

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

A note on the bounds of the error of Gauss Turán-type quadratures Journal of Computational and Applied Mathematic 2 27 276 282 www.elevier.com/locate/cam A note on the bound of the error of Gau Turán-type quadrature Gradimir V. Milovanović a, Miodrag M. Spalević b, a

More information

A trigonometric orthogonality with respect to a nonnegative Borel measure

A trigonometric orthogonality with respect to a nonnegative Borel measure Filomat 6:4 01), 689 696 DOI 10.98/FIL104689M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A trigonometric orthogonality with

More information

Banach Journal of Mathematical Analysis ISSN: (electronic)

Banach Journal of Mathematical Analysis ISSN: (electronic) Banach J. Math. Anal. 1 (7), no. 1, 85 9 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) http://www.math-analysis.org A SPECIAL GAUSSIAN RULE FOR TRIGONOMETRIC POLYNOMIALS GRADIMIR

More information

Gaussian interval quadrature rule for exponential weights

Gaussian interval quadrature rule for exponential weights Gaussian interval quadrature rule for exponential weights Aleksandar S. Cvetković, a, Gradimir V. Milovanović b a Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice

More information

GENERALIZED GAUSS RADAU AND GAUSS LOBATTO FORMULAE

GENERALIZED GAUSS RADAU AND GAUSS LOBATTO FORMULAE BIT 0006-3835/98/3804-0101 $12.00 2000, Vol., No., pp. 1 14 c Swets & Zeitlinger GENERALIZED GAUSS RADAU AND GAUSS LOBATTO FORMULAE WALTER GAUTSCHI 1 1 Department of Computer Sciences, Purdue University,

More information

Applicable Analysis and Discrete Mathematics available online at

Applicable Analysis and Discrete Mathematics available online at Applicable Analysis and Discrete Mathematics available online at http://pefmath.etf.rs Appl. Anal. Discrete Math. x (xxxx), xxx xxx. https://doi.org/10.2298/aadm180730018m QUADATUE WITH MULTIPLE NODES,

More information

Gauss Hermite interval quadrature rule

Gauss Hermite interval quadrature rule Computers and Mathematics with Applications 54 (2007) 544 555 www.elsevier.com/locate/camwa Gauss Hermite interval quadrature rule Gradimir V. Milovanović, Alesandar S. Cvetović Department of Mathematics,

More information

Bulletin T.CXLV de l Académie serbe des sciences et des arts 2013 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 38

Bulletin T.CXLV de l Académie serbe des sciences et des arts 2013 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 38 Bulletin T.CXLV de l Académie serbe des sciences et des arts 213 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 38 FAMILIES OF EULER-MACLAURIN FORMULAE FOR COMPOSITE GAUSS-LEGENDRE

More information

Electronic Transactions on Numerical Analysis Volume 50, 2018

Electronic Transactions on Numerical Analysis Volume 50, 2018 Electronic Transactions on Numerical Analysis Volume 50, 2018 Contents 1 The Lanczos algorithm and complex Gauss quadrature. Stefano Pozza, Miroslav S. Pranić, and Zdeněk Strakoš. Gauss quadrature can

More information

ZERO DISTRIBUTION OF POLYNOMIALS ORTHOGONAL ON THE RADIAL RAYS IN THE COMPLEX PLANE* G. V. Milovanović, P. M. Rajković and Z. M.

ZERO DISTRIBUTION OF POLYNOMIALS ORTHOGONAL ON THE RADIAL RAYS IN THE COMPLEX PLANE* G. V. Milovanović, P. M. Rajković and Z. M. FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. 12 (1997), 127 142 ZERO DISTRIBUTION OF POLYNOMIALS ORTHOGONAL ON THE RADIAL RAYS IN THE COMPLEX PLANE* G. V. Milovanović, P. M. Rajković and Z. M. Marjanović

More information

Interpolation and Cubature at Geronimus Nodes Generated by Different Geronimus Polynomials

Interpolation and Cubature at Geronimus Nodes Generated by Different Geronimus Polynomials Interpolation and Cubature at Geronimus Nodes Generated by Different Geronimus Polynomials Lawrence A. Harris Abstract. We extend the definition of Geronimus nodes to include pairs of real numbers where

More information

Numerical quadratures and orthogonal polynomials

Numerical quadratures and orthogonal polynomials Stud. Univ. Babeş-Bolyai Math. 56(2011), No. 2, 449 464 Numerical quadratures and orthogonal polynomials Gradimir V. Milovanović Abstract. Orthogonal polynomials of different kinds as the basic tools play

More information

The Hardy-Littlewood Function: An Exercise in Slowly Convergent Series

The Hardy-Littlewood Function: An Exercise in Slowly Convergent Series The Hardy-Littlewood Function: An Exercise in Slowly Convergent Series Walter Gautschi Department of Computer Sciences Purdue University West Lafayette, IN 4797-66 U.S.A. Dedicated to Olav Njåstad on the

More information

On the solution of integral equations of the first kind with singular kernels of Cauchy-type

On the solution of integral equations of the first kind with singular kernels of Cauchy-type International Journal of Mathematics and Computer Science, 7(202), no. 2, 29 40 M CS On the solution of integral equations of the first kind with singular kernels of Cauchy-type G. E. Okecha, C. E. Onwukwe

More information

27r---./p. IR.(f)l<=,, l(o) max If(z)l. (1.2) o={z" z= 1/2(u + u-1), u p ei, O<- O <-- 2.a }

27r---./p. IR.(f)l<=,, l(o) max If(z)l. (1.2) o={z z= 1/2(u + u-1), u p ei, O<- O <-- 2.a } SIAM J. NUMER. ANAL. Vol. 27, No. 1, pp. 219-224, February 1990 1990 Society for Industrial and Applied Mathematics 015 A NOTE ON THE CONTOUR INTEGRAL REPRESENTATION OF THE REMAINDER TERM FOR A GAUSS-CHEBYSHEV

More information

Instructions for Matlab Routines

Instructions for Matlab Routines Instructions for Matlab Routines August, 2011 2 A. Introduction This note is devoted to some instructions to the Matlab routines for the fundamental spectral algorithms presented in the book: Jie Shen,

More information

Quadrature Rules With an Even Number of Multiple Nodes and a Maximal Trigonometric Degree of Exactness

Quadrature Rules With an Even Number of Multiple Nodes and a Maximal Trigonometric Degree of Exactness Filomat 9:10 015), 39 55 DOI 10.98/FIL151039T Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Quadrature Rules With an Even Number

More information

Positive definite solutions of some matrix equations

Positive definite solutions of some matrix equations Available online at www.sciencedirect.com Linear Algebra and its Applications 49 (008) 40 44 www.elsevier.com/locate/laa Positive definite solutions of some matrix equations Aleksandar S. Cvetković, Gradimir

More information

Error estimates for Gauss Turán quadratures and their Kronrod extensions

Error estimates for Gauss Turán quadratures and their Kronrod extensions IMA Journal of Numerical Analyi (29 29, 486 57 doi:.93/imanum/drm4 Advance Acce publication on May 6, 28 Error etimate for Gau Turán quadrature and their Kronrod extenion GRADIMIR V. MILOVANOVIĆ Faculty

More information

Contemporary Mathematicians

Contemporary Mathematicians Contemporary Mathematicians Joseph P.S. Kung University of North Texas, USA Editor For further volumes: http://www.springer.com/series/4817 Claude Brezinski Ahmed Sameh Editors Walter Gautschi, Volume

More information

On summation/integration methods for slowly convergent series

On summation/integration methods for slowly convergent series Stud. Univ. Babeş-Bolyai Math. 6(26), o. 3, 359 375 On summation/integration methods for slowly convergent series Gradimir V. Milovanović Dedicated to Professor Gheorghe Coman on the occasion of his 8th

More information

INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS AND THE ERROR TERM OF INTERPOLATORY QUADRATURE FORMULAE FOR ANALYTIC FUNCTIONS

INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS AND THE ERROR TERM OF INTERPOLATORY QUADRATURE FORMULAE FOR ANALYTIC FUNCTIONS MATHEMATICS OF COMPUTATION Volume 75 Number 255 July 2006 Pages 27 23 S 0025-578(06)0859-X Article electronically published on May 2006 INTEGRAL FORMULAS FOR CHEBYSHEV POLYNOMIALS AND THE ERROR TERM OF

More information

Asymptotics of Integrals of. Hermite Polynomials

Asymptotics of Integrals of. Hermite Polynomials Applied Mathematical Sciences, Vol. 4, 010, no. 61, 04-056 Asymptotics of Integrals of Hermite Polynomials R. B. Paris Division of Complex Systems University of Abertay Dundee Dundee DD1 1HG, UK R.Paris@abertay.ac.uk

More information

On Tricomi and Hermite-Tricomi Matrix Functions of Complex Variable

On Tricomi and Hermite-Tricomi Matrix Functions of Complex Variable Communications in Mathematics and Applications Volume (0), Numbers -3, pp. 97 09 RGN Publications http://www.rgnpublications.com On Tricomi and Hermite-Tricomi Matrix Functions of Complex Variable A. Shehata

More information

A quadrature rule of interpolatory type for Cauchy integrals

A quadrature rule of interpolatory type for Cauchy integrals Journal of Computational and Applied Mathematics 126 (2000) 207 220 www.elsevier.nl/locate/cam A quadrature rule of interpolatory type for Cauchy integrals Philsu Kim, U. Jin Choi 1 Department of Mathematics,

More information

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n.

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n. Scientiae Mathematicae Japonicae Online, e-014, 145 15 145 A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS Masaru Nagisa Received May 19, 014 ; revised April 10, 014 Abstract. Let f be oeprator monotone

More information

GAUSS-KRONROD QUADRATURE FORMULAE A SURVEY OF FIFTY YEARS OF RESEARCH

GAUSS-KRONROD QUADRATURE FORMULAE A SURVEY OF FIFTY YEARS OF RESEARCH Electronic Transactions on Numerical Analysis. Volume 45, pp. 371 404, 2016. Copyright c 2016,. ISSN 1068 9613. ETNA GAUSS-KRONROD QUADRATURE FORMULAE A SURVEY OF FIFTY YEARS OF RESEARCH SOTIRIOS E. NOTARIS

More information

SHARP BOUNDS FOR THE GENERAL RANDIĆ INDEX R 1 OF A GRAPH

SHARP BOUNDS FOR THE GENERAL RANDIĆ INDEX R 1 OF A GRAPH ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number, 207 SHARP BOUNDS FOR THE GENERAL RANDIĆ INDEX R OF A GRAPH EI MILOVANOVIĆ, PM BEKAKOS, MP BEKAKOS AND IŽ MILOVANOVIĆ ABSTRACT Let G be an undirected

More information

Polynomial expansions in the Borel region

Polynomial expansions in the Borel region Polynomial expansions in the Borel region By J. T. HURT and L. R. FORD, Rice Institute, Houston, Texas. (Received 22nd August, 1936. Read 6th November, 1936.) This paper deals with certain expansions of

More information

Summation of Series and Gaussian Quadratures

Summation of Series and Gaussian Quadratures Summation of Series Gaussian Quadratures GRADIMIR V. MILOVANOVIĆ Dedicated to Walter Gautschi on the occasion of his 65th birthday Abstract. In 985, Gautschi the author constructed Gaussian quadrature

More information

Part II NUMERICAL MATHEMATICS

Part II NUMERICAL MATHEMATICS Part II NUMERICAL MATHEMATICS BIT 31 (1991). 438-446. QUADRATURE FORMULAE ON HALF-INFINITE INTERVALS* WALTER GAUTSCHI Department of Computer Sciences, Purdue University, West Lafayette, IN 47907, USA Abstract.

More information

ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT. Gradimir V. Milovanović, Aleksandar S. Cvetković, and Zvezdan M.

ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT. Gradimir V. Milovanović, Aleksandar S. Cvetković, and Zvezdan M. PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 8498 2008, 49 60 DOI: 02298/PIM0898049M ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT Gradimir V Milovanović, Aleksandar S Cvetković,

More information

ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT

ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT Received: 31 July, 2008 Accepted: 06 February, 2009 Communicated by: SIMON J SMITH Department of Mathematics and Statistics La Trobe University,

More information

A mean value theorem for systems of integrals

A mean value theorem for systems of integrals J. Math. Anal. Appl. 342 (2008) 334 339 www.elsevier.com/locate/jmaa A mean value theorem for systems of integrals Slobodanka Janković a, Milan Merkle b, a Mathematical nstitute SANU, Knez Mihailova 35,

More information

CS 450 Numerical Analysis. Chapter 8: Numerical Integration and Differentiation

CS 450 Numerical Analysis. Chapter 8: Numerical Integration and Differentiation Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80

More information

(1-2) fx\-*f(x)dx = ^^ Z f(x[n)) + R (f), a > - 1,

(1-2) fx\-*f(x)dx = ^^ Z f(x[n)) + R (f), a > - 1, MATHEMATICS OF COMPUTATION, VOLUME 29, NUMBER 129 JANUARY 1975, PAGES 93-99 Nonexistence of Chebyshev-Type Quadratures on Infinite Intervals* By Walter Gautschi Dedicated to D. H. Lehmer on his 10th birthday

More information

On orthogonal polynomials for certain non-definite linear functionals

On orthogonal polynomials for certain non-definite linear functionals On orthogonal polynomials for certain non-definite linear functionals Sven Ehrich a a GSF Research Center, Institute of Biomathematics and Biometry, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany Abstract

More information

Intrinsic Definition of Strong Shape for Compact Metric Spaces

Intrinsic Definition of Strong Shape for Compact Metric Spaces Volume 39, 0 Pages 7 39 http://topology.auburn.edu/tp/ Intrinsic Definition of Strong Shape for Compact Metric Spaces by Nikita Shekutkovski Electronically published on April 4, 0 Topology Proceedings

More information

COMPUTATION OF BESSEL AND AIRY FUNCTIONS AND OF RELATED GAUSSIAN QUADRATURE FORMULAE

COMPUTATION OF BESSEL AND AIRY FUNCTIONS AND OF RELATED GAUSSIAN QUADRATURE FORMULAE BIT 6-85//41-11 $16., Vol. 4, No. 1, pp. 11 118 c Swets & Zeitlinger COMPUTATION OF BESSEL AND AIRY FUNCTIONS AND OF RELATED GAUSSIAN QUADRATURE FORMULAE WALTER GAUTSCHI Department of Computer Sciences,

More information

Applicable Analysis and Discrete Mathematics available online at

Applicable Analysis and Discrete Mathematics available online at Applicable Analysis and Discrete Mathematics available online at http://pefmath.etf.rs Appl. Anal. Discrete Math. x (xxxx, xxx xxx. doi:.98/aadmxxxxxxxx A STUDY OF GENERALIZED SUMMATION THEOREMS FOR THE

More information

Bivariate Lagrange interpolation at the Padua points: The generating curve approach

Bivariate Lagrange interpolation at the Padua points: The generating curve approach Journal of Approximation Theory 43 (6) 5 5 www.elsevier.com/locate/jat Bivariate Lagrange interpolation at the Padua points: The generating curve approach Len Bos a, Marco Caliari b, Stefano De Marchi

More information

On spectral radius and energy of complete multipartite graphs

On spectral radius and energy of complete multipartite graphs Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn., ISSN 1855-3974 (electronic edn. ARS MATHEMATICA CONTEMPORANEA 9 (2015 109 113 On spectral radius and energy of complete multipartite

More information

Numerical Methods I: Numerical Integration/Quadrature

Numerical Methods I: Numerical Integration/Quadrature 1/20 Numerical Methods I: Numerical Integration/Quadrature Georg Stadler Courant Institute, NYU stadler@cims.nyu.edu November 30, 2017 umerical integration 2/20 We want to approximate the definite integral

More information

Quadrature rules with multiple nodes

Quadrature rules with multiple nodes Quadrature rules with multiple nodes Gradimir V. Milovanović and Marija P. Stanić Abstract In this paper a brief historical survey of the development of quadrature rules with multiple nodes and the maximal

More information

Bulletin T.CXXII de l Académie Serbe des Sciences et des Arts Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 26

Bulletin T.CXXII de l Académie Serbe des Sciences et des Arts Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 26 Bulletin T.CXXII de l Académie Serbe des Sciences et des Arts - 001 Classe des Sciences mathématiques et naturelles Sciences mathématiques, No 6 SOME SPECTRAL PROPERTIES OF STARLIKE TREES M. LEPOVIĆ, I.

More information

MATH 117 LECTURE NOTES

MATH 117 LECTURE NOTES MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set

More information

Fully Symmetric Interpolatory Rules for Multiple Integrals over Infinite Regions with Gaussian Weight

Fully Symmetric Interpolatory Rules for Multiple Integrals over Infinite Regions with Gaussian Weight Fully Symmetric Interpolatory ules for Multiple Integrals over Infinite egions with Gaussian Weight Alan Genz Department of Mathematics Washington State University ullman, WA 99164-3113 USA genz@gauss.math.wsu.edu

More information

COMPUTATION OF GAUSS-KRONROD QUADRATURE RULES

COMPUTATION OF GAUSS-KRONROD QUADRATURE RULES MATHEMATICS OF COMPUTATION Volume 69, Number 231, Pages 1035 1052 S 0025-5718(00)01174-1 Article electronically published on February 17, 2000 COMPUTATION OF GAUSS-KRONROD QUADRATURE RULES D.CALVETTI,G.H.GOLUB,W.B.GRAGG,ANDL.REICHEL

More information

Discrete Orthogonal Polynomials on Equidistant Nodes

Discrete Orthogonal Polynomials on Equidistant Nodes International Mathematical Forum, 2, 2007, no. 21, 1007-1020 Discrete Orthogonal Polynomials on Equidistant Nodes Alfredo Eisinberg and Giuseppe Fedele Dip. di Elettronica, Informatica e Sistemistica Università

More information

ON A SINE POLYNOMIAL OF TURÁN

ON A SINE POLYNOMIAL OF TURÁN ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 48, Number 1, 18 ON A SINE POLYNOMIAL OF TURÁN HORST ALZER AND MAN KAM KWONG ABSTRACT. In 1935, Turán proved that ( n + a j ) S n,a() = sin(j) >, n j n, a N,

More information

In numerical analysis quadrature refers to the computation of definite integrals.

In numerical analysis quadrature refers to the computation of definite integrals. Numerical Quadrature In numerical analysis quadrature refers to the computation of definite integrals. f(x) a x i x i+1 x i+2 b x A traditional way to perform numerical integration is to take a piece of

More information

Quadrature Formula for Computed Tomography

Quadrature Formula for Computed Tomography Quadrature Formula for Computed Tomography orislav ojanov, Guergana Petrova August 13, 009 Abstract We give a bivariate analog of the Micchelli-Rivlin quadrature for computing the integral of a function

More information

SZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART

SZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART Georgian Mathematical Journal Volume 4 (27), Number 4, 673 68 SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS ON HE SEGMEN [, +]: HE CASE OF A MEASURE WIH FINIE DISCREE PAR RABAH KHALDI Abstract. he strong asymptotics

More information

Review of Power Series

Review of Power Series Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power

More information

Orthogonal Polynomials, Quadratures & Sparse-Grid Methods for Probability Integrals

Orthogonal Polynomials, Quadratures & Sparse-Grid Methods for Probability Integrals 1/31 Orthogonal Polynomials, Quadratures & Sparse-Grid Methods for Probability Integrals Dr. Abebe Geletu May, 2010 Technische Universität Ilmenau, Institut für Automatisierungs- und Systemtechnik Fachgebiet

More information

A collocation method for solving the fractional calculus of variation problems

A collocation method for solving the fractional calculus of variation problems Bol. Soc. Paran. Mat. (3s.) v. 35 1 (2017): 163 172. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v35i1.26333 A collocation method for solving the fractional

More information

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by D. Let

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by D. Let HOW FAR IS AN ULTRAFLAT SEQUENCE OF UNIMODULAR POLYNOMIALS FROM BEING CONJUGATE-RECIPROCAL? Tamás Erdélyi Abstract. In this paper we study ultraflat sequences (P n) of unimodular polynomials P n K n in

More information

A Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions

A Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions mathematics of computation volume 39, number 159 july 1982, pages 201-206 A Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions By John P. Boyd Abstract. The theorem proved here extends

More information

Quadratures and integral transforms arising from generating functions

Quadratures and integral transforms arising from generating functions Quadratures and integral transforms arising from generating functions Rafael G. Campos Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana, 58060, Morelia, México. rcampos@umich.mx Francisco

More information

UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS

UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS J. Korean Math. Soc. 45 (008), No. 3, pp. 835 840 UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS Paulius Drungilas Reprinted from the Journal of the Korean Mathematical Society Vol. 45, No. 3, May

More information

A Note about the Pochhammer Symbol

A Note about the Pochhammer Symbol Mathematica Moravica Vol. 12-1 (2008), 37 42 A Note about the Pochhammer Symbol Aleksandar Petoević Abstract. In this paper we give elementary proofs of the generating functions for the Pochhammer symbol

More information

NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS

NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS Tamás Erdélyi Dedicated to the memory of George G Lorentz Abstract Let Λ n := {λ 0 < λ < < λ n } be a set of real numbers The collection of all linear

More information

The Generating Functions for Pochhammer

The Generating Functions for Pochhammer The Generating Functions for Pochhammer Symbol { }, n N Aleksandar Petoević University of Novi Sad Teacher Training Faculty, Department of Mathematics Podgorička 4, 25000 Sombor SERBIA and MONTENEGRO Email

More information

Szegő-Lobatto quadrature rules

Szegő-Lobatto quadrature rules Szegő-Lobatto quadrature rules Carl Jagels a,, Lothar Reichel b,1, a Department of Mathematics and Computer Science, Hanover College, Hanover, IN 47243, USA b Department of Mathematical Sciences, Kent

More information

SQUARE-ROOT FAMILIES FOR THE SIMULTANEOUS APPROXIMATION OF POLYNOMIAL MULTIPLE ZEROS

SQUARE-ROOT FAMILIES FOR THE SIMULTANEOUS APPROXIMATION OF POLYNOMIAL MULTIPLE ZEROS Novi Sad J. Math. Vol. 35, No. 1, 2005, 59-70 SQUARE-ROOT FAMILIES FOR THE SIMULTANEOUS APPROXIMATION OF POLYNOMIAL MULTIPLE ZEROS Lidija Z. Rančić 1, Miodrag S. Petković 2 Abstract. One-parameter families

More information

POLYNOMIAL INTERPOLATION IN NONDIVISION ALGEBRAS

POLYNOMIAL INTERPOLATION IN NONDIVISION ALGEBRAS Electronic Transactions on Numerical Analysis. Volume 44, pp. 660 670, 2015. Copyright c 2015,. ISSN 1068 9613. ETNA POLYNOMIAL INTERPOLATION IN NONDIVISION ALGEBRAS GERHARD OPFER Abstract. Algorithms

More information

Positivity of Turán determinants for orthogonal polynomials

Positivity of Turán determinants for orthogonal polynomials Positivity of Turán determinants for orthogonal polynomials Ryszard Szwarc Abstract The orthogonal polynomials p n satisfy Turán s inequality if p 2 n (x) p n 1 (x)p n+1 (x) 0 for n 1 and for all x in

More information

Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation

Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation Journal of Computational and Applied Mathematics 127 (21) 267 286 www.elsevier.nl/locate/cam Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation Gradimir V.

More information

A Fixed Point Theorem for G-Monotone Multivalued Mapping with Application to Nonlinear Integral Equations

A Fixed Point Theorem for G-Monotone Multivalued Mapping with Application to Nonlinear Integral Equations Filomat 31:7 (217), 245 252 DOI 1.2298/FIL17745K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Fixed Point Theorem for G-Monotone

More information

On Some Mean Value Results for the Zeta-Function and a Divisor Problem

On Some Mean Value Results for the Zeta-Function and a Divisor Problem Filomat 3:8 (26), 235 2327 DOI.2298/FIL6835I Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Some Mean Value Results for the

More information

x x2 2 + x3 3 x4 3. Use the divided-difference method to find a polynomial of least degree that fits the values shown: (b)

x x2 2 + x3 3 x4 3. Use the divided-difference method to find a polynomial of least degree that fits the values shown: (b) Numerical Methods - PROBLEMS. The Taylor series, about the origin, for log( + x) is x x2 2 + x3 3 x4 4 + Find an upper bound on the magnitude of the truncation error on the interval x.5 when log( + x)

More information

9. Series representation for analytic functions

9. Series representation for analytic functions 9. Series representation for analytic functions 9.. Power series. Definition: A power series is the formal expression S(z) := c n (z a) n, a, c i, i =,,, fixed, z C. () The n.th partial sum S n (z) is

More information

Orthogonal polynomials

Orthogonal polynomials Orthogonal polynomials Gérard MEURANT October, 2008 1 Definition 2 Moments 3 Existence 4 Three-term recurrences 5 Jacobi matrices 6 Christoffel-Darboux relation 7 Examples of orthogonal polynomials 8 Variable-signed

More information

AP Calculus Testbank (Chapter 9) (Mr. Surowski)

AP Calculus Testbank (Chapter 9) (Mr. Surowski) AP Calculus Testbank (Chapter 9) (Mr. Surowski) Part I. Multiple-Choice Questions n 1 1. The series will converge, provided that n 1+p + n + 1 (A) p > 1 (B) p > 2 (C) p >.5 (D) p 0 2. The series

More information

Invariant Approximation Results of Generalized Contractive Mappings

Invariant Approximation Results of Generalized Contractive Mappings Filomat 30:14 (016), 3875 3883 DOI 10.98/FIL1614875C Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Invariant Approximation Results

More information

A Proof of the Lucas-Lehmer Test and its Variations by Using a Singular Cubic Curve

A Proof of the Lucas-Lehmer Test and its Variations by Using a Singular Cubic Curve 1 47 6 11 Journal of Integer Sequences, Vol. 1 (018), Article 18.6. A Proof of the Lucas-Lehmer Test and its Variations by Using a Singular Cubic Curve Ömer Küçüksakallı Mathematics Department Middle East

More information

CONSTRUCTIVE APPROXIMATION 2001 Springer-Verlag New York Inc.

CONSTRUCTIVE APPROXIMATION 2001 Springer-Verlag New York Inc. Constr. Approx. (2001) 17: 267 274 DOI: 10.1007/s003650010021 CONSTRUCTIVE APPROXIMATION 2001 Springer-Verlag New York Inc. Faber Polynomials Corresponding to Rational Exterior Mapping Functions J. Liesen

More information

Representation for the generalized Drazin inverse of block matrices in Banach algebras

Representation for the generalized Drazin inverse of block matrices in Banach algebras Representation for the generalized Drazin inverse of block matrices in Banach algebras Dijana Mosić and Dragan S. Djordjević Abstract Several representations of the generalized Drazin inverse of a block

More information

ON LINEAR COMBINATIONS OF

ON LINEAR COMBINATIONS OF Física Teórica, Julio Abad, 1 7 (2008) ON LINEAR COMBINATIONS OF ORTHOGONAL POLYNOMIALS Manuel Alfaro, Ana Peña and María Luisa Rezola Departamento de Matemáticas and IUMA, Facultad de Ciencias, Universidad

More information

Convergence and Stability of a New Quadrature Rule for Evaluating Hilbert Transform

Convergence and Stability of a New Quadrature Rule for Evaluating Hilbert Transform Convergence and Stability of a New Quadrature Rule for Evaluating Hilbert Transform Maria Rosaria Capobianco CNR - National Research Council of Italy Institute for Computational Applications Mauro Picone,

More information

CESÁRO TYPE OPERATORS ON SPACES OF ANALYTIC FUNCTIONS. S. Naik

CESÁRO TYPE OPERATORS ON SPACES OF ANALYTIC FUNCTIONS. S. Naik Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 25:4 2, 85 97 DOI:.2298/FIL485N CESÁRO TYPE OPERATORS ON SPACES OF ANALYTIC FUNCTIONS

More information

UNIFORM BOUNDS FOR BESSEL FUNCTIONS

UNIFORM BOUNDS FOR BESSEL FUNCTIONS Journal of Applied Analysis Vol. 1, No. 1 (006), pp. 83 91 UNIFORM BOUNDS FOR BESSEL FUNCTIONS I. KRASIKOV Received October 8, 001 and, in revised form, July 6, 004 Abstract. For ν > 1/ and x real we shall

More information

ESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION

ESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION Manuscript submitted to Website: http://aimsciences.org AIMS Journals Volume 00, Number 0, Xxxx XXXX pp. 000 000 ESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION NAEEM M.H. ALKOUMI

More information

Numerical Integration (Quadrature) Another application for our interpolation tools!

Numerical Integration (Quadrature) Another application for our interpolation tools! Numerical Integration (Quadrature) Another application for our interpolation tools! Integration: Area under a curve Curve = data or function Integrating data Finite number of data points spacing specified

More information

c 2007 Society for Industrial and Applied Mathematics

c 2007 Society for Industrial and Applied Mathematics SIAM J. SCI. COMPUT. Vol. 9, No. 3, pp. 07 6 c 007 Society for Industrial and Applied Mathematics NUMERICAL QUADRATURES FOR SINGULAR AND HYPERSINGULAR INTEGRALS IN BOUNDARY ELEMENT METHODS MICHAEL CARLEY

More information

A note on a construction of J. F. Feinstein

A note on a construction of J. F. Feinstein STUDIA MATHEMATICA 169 (1) (2005) A note on a construction of J. F. Feinstein by M. J. Heath (Nottingham) Abstract. In [6] J. F. Feinstein constructed a compact plane set X such that R(X), the uniform

More information

Math Subject GRE Questions

Math Subject GRE Questions Math Subject GRE Questions Calculus and Differential Equations 1. If f() = e e, then [f ()] 2 [f()] 2 equals (a) 4 (b) 4e 2 (c) 2e (d) 2 (e) 2e 2. An integrating factor for the ordinary differential equation

More information

arxiv: v1 [math.mg] 15 Jul 2013

arxiv: v1 [math.mg] 15 Jul 2013 INVERSE BERNSTEIN INEQUALITIES AND MIN-MAX-MIN PROBLEMS ON THE UNIT CIRCLE arxiv:1307.4056v1 [math.mg] 15 Jul 013 TAMÁS ERDÉLYI, DOUGLAS P. HARDIN, AND EDWARD B. SAFF Abstract. We give a short and elementary

More information

Applied Mathematics and Computation

Applied Mathematics and Computation Alied Mathematics and Comutation 18 (011) 944 948 Contents lists available at ScienceDirect Alied Mathematics and Comutation journal homeage: www.elsevier.com/locate/amc A generalized Birkhoff Young Chebyshev

More information

Tools from Lebesgue integration

Tools from Lebesgue integration Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given

More information

Complex Series. Chapter 20

Complex Series. Chapter 20 hapter 20 omplex Series As in the real case, representation of functions by infinite series of simpler functions is an endeavor worthy of our serious consideration. We start with an examination of the

More information

On Fixed Point Results for Matkowski Type of Mappings in G-Metric Spaces

On Fixed Point Results for Matkowski Type of Mappings in G-Metric Spaces Filomat 29:10 2015, 2301 2309 DOI 10.2298/FIL1510301G Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Fixed Point Results for

More information

ASYMPTOTIC BEHAVIOUR OF SECOND-ORDER DIFFERENCE EQUATIONS

ASYMPTOTIC BEHAVIOUR OF SECOND-ORDER DIFFERENCE EQUATIONS ANZIAM J. 462004, 57 70 ASYMPTOTIC BEHAVIOUR OF SECOND-ORDER DIFFERENCE EQUATIONS STEVO STEVIĆ Received 9 December, 200; revised 9 September, 2003 Abstract In this paper we prove several growth theorems

More information

3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?

3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure? MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable

More information

Random Bernstein-Markov factors

Random Bernstein-Markov factors Random Bernstein-Markov factors Igor Pritsker and Koushik Ramachandran October 20, 208 Abstract For a polynomial P n of degree n, Bernstein s inequality states that P n n P n for all L p norms on the unit

More information

Complex Analysis Slide 9: Power Series

Complex Analysis Slide 9: Power Series Complex Analysis Slide 9: Power Series MA201 Mathematics III Department of Mathematics IIT Guwahati August 2015 Complex Analysis Slide 9: Power Series 1 / 37 Learning Outcome of this Lecture We learn Sequence

More information

arxiv: v1 [cs.ds] 11 Oct 2018

arxiv: v1 [cs.ds] 11 Oct 2018 Path matrix and path energy of graphs arxiv:1810.04870v1 [cs.ds] 11 Oct 018 Aleksandar Ilić Facebook Inc, Menlo Park, California, USA e-mail: aleksandari@gmail.com Milan Bašić Faculty of Sciences and Mathematics,

More information

POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET

POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9947(XX)0000-0 POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET PETER BORWEIN, TAMÁS ERDÉLYI, FRIEDRICH LITTMANN

More information