Banach Journal of Mathematical Analysis ISSN: (electronic)

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1 Banach J. Math. Anal. 1 (7), no. 1, 85 9 Banach Journal of Mathematical Analysis ISSN: (electronic) A SPECIAL GAUSSIAN RULE FOR TRIGONOMETRIC POLYNOMIALS GRADIMIR V. MILOVANOVIĆ1, ALEKSANDAR S. CVETKOVIĆ AND MARIJA P. STANIĆ3 This paper is dedicated to Professor Themistocles M. Rassias. Submitted by S. S. Dragomir Abstract. Abram Haimovich Turetzkii [Uchenye Zapiski, 1 (149) (1959), (translation in English in East J. Approx. 11 (5), )] considered interpolatory uadrature rules which have the following form f(x)w(x)dx n ν= w νf(x ν ), and which are exact for all trigonometric polynomials of degree less than or eual to n. Maximal trigonometric degree of exactness of such uadratures is n, and such kind of uadratures are known as uadratures of Gaussian type or Gaussian uadratures for trigonometric polynomials. In this paper we prove some interesting properties of a special Gaussian uadrature with respect to the weight function w m (x) = 1 + sin mx, where m is a positive integer. 1. Introduction This paper is dedicated to Themistocles M. Rassias for our long and fruitful collaboration in mathematical research, including the writing of our book [4], published in 1994 by World Scientific Publishing Co., jointly with Professor Dragoslav S. Mitrinović. Let the weight function w(x) be integrable and nonnegative on the interval [, π), vanishing there only on a set of a measure zero. Date: Received: 4 March 7; Accepted: 4 October 7. Corresponding author. Mathematics Subject Classification. Primary 65D3; Secondary 4A5. Key words and phrases. Trigonometric polynomials of semi-integer degree, Orthogonality, Gaussian type uadratures. 85

2 86 G.V. MILOVANOVIĆ, A.S. CVETKOVIĆ, M.P. STANIĆ By T n we denote the linear space of all trigonometric polynomials of degree less than or eual to n, and by Tn 1/ we denote the linear span of the following set { cos(ν + 1/)x, sin(ν + 1/)x, ν =, 1,..., n }. Trigonometric functions from T 1/ n, i.e., trigonometric functions of the following form n [ ( c ν cos ν + 1 ) ( x + d ν sin ν + 1 ) x ν= ], (1.1) where c ν, d ν R, c n + d n =, are known as trigonometric polynomials of semi integer degree n + 1/ (see [5], []). This kind of trigonometric functions are important for construction of the Gaussian uadrature rules for trigonometric polynomials. It is obvious that A n+1/ (x) = A n k= sin x x k (A is a non-zero constant), (1.) is trigonometric polynomial of semi integer degree n + 1/. Also, every trigonometric polynomial of semi-integer degree n + 1/ of the form (1.1) can be represented in the form (1.) (see [5, Lemma 1]). For any positive integer n, the uadrature rule of the Gaussian type is the following one where weights w ν are given by t(x)w(x)dx = n ν= w ν t(x ν ), t T n, (1.3) A n+1/ (x) w ν = sin x x w(x)dx, ν =, 1,..., n, (1.4) ν A n+1/ (x ν) and nodes are zeros of A n+1/ (x), which is orthogonal on [, π) with respect to the weight function w(x) to every trigonometric polynomial of the semi-integer degree from T 1/ n 1 (see [5], []). It is known that such orthogonal trigonometric polynomial of semi integer degree A n+1/ (x) has in [, π) exactly n + 1 distinct simple zeros (see [5, Theorem 3]) and A n+1/ with given leading coefficients c n and d n is uniuely determined (see [5, 3.]). In [] and [3] two choices of leading coefficients were considered. For the choice c n = 1, d n =, we denote an orthogonal trigonometric polynomial of semi-integer degree by A C n+1/, and for the choice c n = and d n = 1 by A S n+1/. The Gaussian uadrature (1.3) is not given uniuely since it is possible to use any orthogonal polynomial of semi-integer degree A n+1/ Tn 1/. In [] a numerical method for constructing the Gaussian uadratures using A C n+1/ is given. That method is based on five term recurrence relations for A C n+1/ and AS n+1/. The main problem in that method is the calculation of five term recurrence coefficients. Explicit formulas for five term recurrence coefficients for some weight functions are given in [3].

3 GAUSSIAN RULE FOR TRIGONOMETRIC POLYNOMIALS 87 In this paper, in Section, we prove some interesting properties for Gaussian uadratures (1.3) with respect to the weight function w m (x) = 1+sin mx, m N, and in Section 3 we give a numerical example.. Main results As an example, the nodes x ν and weight w ν, ν =, 1,..., n, for the Gaussian uadrature for n = 5 and w 15 (x) = 1 + sin 15x are given in []. In that case we have seen some kind of symmetry, since w ν = w ν+17j and x ν+17j = x ν + jπ/3, j = 1,, ν =, 1,..., 16. This symmetry is not an isolated case, namely Gaussian uadratures (1.3) with respect to the weight functions w m (x) = 1+sin mx, m N, have interesting properties, which are presented in Theorem.1. If we want to construct a Gaussian uadrature rule with n nodes for the weight function w m1 (x) = 1 + sin m 1 x where gcd(m 1, n 1 + 1) = d 1 we can obtain nodes and weights for such uadrature directly from nodes and weights of the Gaussian uadrature rule with (n 1 + 1)/d nodes with respect to the weight w m1 /d. In a given example in [] we have n 1 = 5, m 1 = 15 and gcd(15, 51) = 3. Theorem.1. Let denote by x ν, w ν, ν = 1,..., n + 1, the nodes and weights for the Gaussian uadrature rule with respect to the weight function w m (x) = 1 + sin mx. Then x j(n+1)+ν = x ν + jπ, ŵ j(n+1)+ν = w ν, for j =, 1,..., 1, ν = 1,..., n + 1, are nodes and weights for the Gaussian uadrature rule with (n+1) nodes with respect to the weight function w m (x) = 1 + sin mx, where is an odd positive integer. Proof. According to orthogonality conditions for trigonometric polynomials of semi integer degree with respect to the weight function w m (x) = 1 + sin mx, we obtain A C n+1/(x) cos(k + 1/)x(1 + sin mx)dx =, k n 1, A C n+1/(x) sin(k + 1/)x(1 + sin mx)dx =, k n 1. If we introduce x := x, we have / / A C n+1/(x) cos(k + 1/)x(1 + sin mx)dx =, k n 1, A C n+1/(x) sin(k + 1/)x(1 + sin mx)dx =, k n 1. Substituting t = x + jπ/, j = 1,..., 1, we get (j+1)π jπ A C n+1/(t jπ) cos(k + 1/)(t jπ)(1 + sin m(t jπ))dt =,

4 88 G.V. MILOVANOVIĆ, A.S. CVETKOVIĆ, M.P. STANIĆ (j+1)π jπ A C n+1/(t jπ) sin(k + 1/)(t jπ)(1 + sin m(t jπ))dt for k n 1, i.e., (j+1)π/ jπ/ (j+1)π/ jπ/ A C n+1/(t) cos(k + 1/)t(1 + sin mt)dt =, A C n+1/(t) sin(k + 1/)t(1 + sin mt)dt =, because of cos(k + 1/)(t jπ) = ( 1) j cos(k + 1/)t and sin(k + 1/)(t jπ) = ( 1) j sin(k + 1/)t, for j = 1,..., 1. Thus, we have =, A C n+1/(x) cos(k + 1/)x(1 + sin mx)dx =, k n 1, A C n+1/(x) sin(k + 1/)x(1 + sin mx)dx =, k n 1. Let denote by T the linear span of the following functions cos(k + 1/)x = cos(k + ( 1)/ + 1/)x, k =, 1,..., n, sin(k + 1/)x = sin(k + ( 1)/ + 1/)x, k =, 1,..., n. Obviously, T T 1/ n+( 1)/. Using integrals cos(ν + 1/)x cos(l + 1/)xw m (x)dx = πδ ν,l, sin(ν + 1/)x sin(l + 1/)xw m (x)dx = πδ ν,l, cos(ν + 1/)x sin(l + 1/)xw m (x)dx = π (δ ν,l m + δ ν,m l 1 δ ν,l+m ), for ν = k + ( 1)/, we see that integrals might not be eual zero only under condition l = i + ( 1)/, i Z. With this argument we obtain the following orthogonality T (T 1/ n+( 1)/ T ) with respect to the inner product (f, ) = f(x)g(x)w m (x)dx. Then, A C n+1/ (x) is orthogonal on [, π) to all trigonometric polynomials of semi integer degree t T 1/ n+( 1)/ 1 with respect to the weight function w m(x).

5 GAUSSIAN RULE FOR TRIGONOMETRIC POLYNOMIALS 89 Since A C n+1/ (x) can be represented as we get n+1 A C n+1/(x) = A sin x x ν, A, ν=1 n+1 A C n+1/(x) = A sin x x ν ν=1 n+1 = A sin (x x ν/), so it is obvious that x j(n+1)+ν, j =, 1,..., 1, ν = 1,..., n + 1, are zeros of A C n+1/ (x). Formulas for weight coefficients ŵ j(n+1)+ν, j =, 1,..., 1, ν = 1,..., n+1, can be easily obtained using (1.4). ν=1 3. Numerical example We determine the nodes and weights for a Gaussian uadrature with 75 nodes and the weight function w 45 (x) = 1 + sin 45x. Since gcd(45, 75) = 15, according to Theorem.1, we can obtain parameters for this Gaussian uadrature directly from the nodes and weights of a Gaussian uadrature with 5 nodes and the weight function w 3 (x) = 1 + sin 3x. The parameters x ν and weight w ν, ν =, 1,..., n, for n = and w(x) = 1 + sin 3x are given in Table 1. (Numbers in parentheses indicate decimal exponents.) We calculate these parameters using a numerical method described in [] with explicit formulas for five-term recurrence coefficients given in [3]. All computations are performed in double precision arithmetic (16 decimal digits mantissa) in Mathematica, using the corresponding software package described in [1]. Now it is easy to obtain the nodes x ν and the weights ŵ ν, ν =, 1,..., 74, of the Gaussian uadrature for the w 45 (x) = 1 + sin 45x. The nodes x ν, ν =, 1,..., 4, and the weights ŵ k, k =, 1,..., 74, are given in Table. The other nodes x k, k = 5, 6,..., 74, are given by x 5j+ν = x ν + jπ, j = 1,..., 14, ν =, 1,..., Table 1. Nodes x ν and weights w ν, ν =, 1,..., 4, for w 3 (x) = 1 + sin 3x ν x ν w ν ( 1) ( 1) ( 1)

6 9 G.V. MILOVANOVIĆ, A.S. CVETKOVIĆ, M.P. STANIĆ Table. Nodes x ν and weights ŵ 5j+ν, ν =, 1,..., 4, j =, 1,..., 14, for n = 37 and w 45 (x) = 1 + sin 45x ν x ν ŵ 5j+ν, j =, 1,..., ( 1) ( 1) ( 1) ( 1) Acknowledgements: The authors were supported in part by the Serbian Ministry of Science (Project: Orthogonal Systems and Applications, grant number #1444G) and the Swiss National Science Foundation (SCOPES Joint Research Project No. IB New Methods for Quadrature ) References 1. A.S. Cvetković and G.V. Milovanović, The Mathematica Package OrthogonalPolynomials, Facta Univ. Ser. Math. Inform. 19 (4), G.V. Milovanović, A.S. Cvetković and M.P. Stanić, Trigonometric orthogonal systems and uadrature formulae, preprint. 3. G.V. Milovanović, A.S. Cvetković and M.P. Stanić, Explicit formulas for five-term recurrence coefficients of orthogonal trigonometric polynomials of semi-integer degree, Appl. Math. Comput. (to appear). 4. G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias, Topics in Polynomials: Extremal Problems, Ineualities, Zeros, World Scientific, Singapore New Jersey London Hong Kong, A.H. Turetzkii, On uadrature formulae that are exact for trigonometric polynomials, East J. Approx. 11 (5), (translation in English from Uchenye Zapiski, Vypusk 1 (149), Seria math. Theory of Functions, Collection of papers, Izdatel stvo Belgosuniversiteta imeni V.I. Lenina, Minsk (1959), 31 54). 1 Department of Mathematics, Faculty of Electronic Engineering, University of Niš, P.O. Box 73, 18 Niš, Serbia. address: grade@junis.ni.ac.yu Department of Mathematics, Faculty of Electronic Engineering, University of Niš, P.O. Box 73, 18 Niš, Serbia. address: aca@elfak.ni.ac.yu 3 Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, P.O. Box 6, 34 Kragujevac, Serbia. address: stanicm@kg.ac.yu