Quadrature rules with multiple nodes

Transcription

1 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 algebraic degree of exactness is given. The natural generalization of such rules are quadrature rules with multiple nodes and the maximal degree of exactness in some functional spaces that are different from the space of algebraic polynomial. For that purpose we present a generalized quadrature rules considered by A. Ghizzeti and A. Ossicini [Quadrature Formulae, Academie Verlag, Berlin, 1970] and apply their ideas in order to obtain quadrature rules with multiple nodes and the maximal trigonometric degree of exactness. Such quadrature rules are characterized by so called s and σ orthogonal trigonometric polynomials. Numerical method for the construction of such quadrature rules are given, as well as numerical the example which illustrate obtained theoretical results. 1 Introduction and preliminaries The most significant discovery in the field of numerical integration in 19th century was made by Carl Friedrich Gauß in His method [14] dramatically improved the earlier Newton s method. Gauss method was enriched by significant contributions of Jacobi [1] and Christoffel [7]. More than hundred years after Gauß published his famous method there appeared the idea of numerical integration involving multiple nodes. Taking any system of n distinct points {τ 1,...,τ n } and n nonnegative integers m 1,...,m n, and starting from the Hermite interpolation formula, L. Chakalov [4] in 1948 obtained the quadrature formula Gradimir V. Milovanović The Serbian Academy of Sciences and Arts, Kneza Mihaila 35, Belgrade, Serbia gvm@mi.sanu.ac.rs Marija P. Stanić Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovića 1, Kragujevac, Serbia stanicm@kg.ac.rs 1

2 Gradimir V. Milovanović and Marija P. Stanić 1 1 f t)dt = n ν=1 [ A0,ν f τ ν ) + A 1,ν f τ ν ) + + A mν 1,ν f m ν 1) τ ν ) ], 1) which is exact for all polynomials of degree at most m m n 1. He gave a method for computing the coefficients A i,ν in 1). Such coefficients are given by A i,ν = 1 1 l i,νt)dt, ν = 1,...,n, i = 0,1,...,m ν 1, where l i,ν t) are the fundamental functions of Hermite interpolation. Taking m 1 = = m n = k in 1), P. Turán [48] in 1950 studied numerical quadratures of the form 1 1 k 1 f t)dt = n i=0 ν=1 A i,ν f i) τ ν ) + R n,k f ), ) Obviously, for any given nodes 1 τ 1 τ n 1 the formula ) can be made exact for f P kn 1 P m, m N 0, is the set of all algebraic polynomials of degree at most m). However, for k = 1 the formula ), i.e., 1 1 f t)dt = n ν=1 A 0,ν f τ ν ) + R n,1 f ) is exact for all polynomials of degree at most n 1 if the nodes τ ν are the zeros of the Legendre polynomial P n it is the well known Gauss Legendre quadrature rule). Because of that it is quite natural to consider whether nodes τ ν can be chosen in such a way that the quadrature formula ) will be exact for all algebraic polynomials of degree less that or equal to k + 1)n 1. Turán [48] showed that the answer is negative for k =, while it is positive for k = 3. He proved that the nodes τ ν, ν = 1,,...,n, should be chosen as the zeros of the monic polynomial πn t) = t n + which minimizes the integral 1 1 [π nt)] 4 dt, where π n t) = t n + a n 1 t n a 1 t + a 0. In the general case, the answer is negative for even, and positive for odd k, when τ ν, ν = 1,,...,n, must be the zeros of the monic polynomial πn minimizing 1 1 [π nt)] k+1 dt. Specially, for k = 1, πn is the monic Legendre polynomial P n. Let us now assume that k = s + 1, s 0. Instead of ), it is also interesting to consider a more general Gauss Turán type quadrature formula R f t)dλt) = s n i=0 ν=1 A i,ν f i) τ ν ) + R n,s f ), 3) where dλt) is a given nonnegative measure on the real line R, with compact or unbounded support, for which all moments µ k = R tk dλt), k = 0,1,..., exist and are finite, and µ 0 > 0. It is known that formula 3) is exact for all polynomials of degree at most s + 1)n 1, i.e., R n,s f ) = 0 for f P s+1)n 1. The nodes τ ν, ν = 1,...,n, in 3) are the zeros of the monic polynomial π n,s t), which minimizes the integral Fa 0,a 1,...,a n 1 ) = [π n t)] s+ dλt), R

3 Quadrature rules with multiple nodes 3 where π n t) = t n + a n 1 t n a 1 t + a 0. This minimization leads to the conditions 1 s + F = [π n t)] s+1 t k dλt) = 0, k = 0,1,...,n 1. 4) a k R These polynomials π n = π n,s are known as s orthogonal or s self associated) polynomials on R with respect to the measure dλt) for more details see [15, 35, 36, 37]. For s = 0 they reduce to the standard orthogonal polynomials and 3) becomes the well known Gauss Christoffel formula. Using some facts about monosplines, Micchelli [] investigated the sign of the Cotes coefficients A i,ν in the Turán quadrature formula. For the numerical methods for the construction of s orthogonal polynomials as well as for the construction of Gauss Turán-type quadrature formulae we refer readers to the articles [3, 4, 17]. A next natural generalization of the Turán quadrature formula 3) is generalization to rules having nodes with different multiplicities. Such rule for dλt) = dt on a,b) was derived independently by L. Chakalov [5, 6] and T. Popoviciu [38]. Important theoretical progress on this subject was made by D.D. Stancu [43, 44] see also [46]). Let us assume that the nodes τ 1 < τ < < τ n, have multiplicities m 1,m,...,m n, respectively a permutation of the multiplicities m 1,m,...,m n, with the nodes held fixed, in general yields to a new quadrature rule). It can be shown that the quadrature formula 1) is exact for all polynomials of degree less than n ν=1 [m ν +1)/]. Thus, the multiplicities m ν that are even do not contribute toward an increase in the degree of exactness, so that it is reasonable to assume that all m ν are odd integers, i.e., that m ν = s ν +1, ν = 1,,...,n. Therefore, for a given sequence of nonnegative integers σ = s 1,s,...,s n ) the corresponding quadrature formula R f t)dλt) = n s ν ν=1 i=0 A i,ν f i) τ ν ) + R f ) has the maximal degree of exactness d max = n ν=1 s ν + n 1 if and only if n R ν=1 t τ ν ) s ν +1 t k dλt) = 0, k = 0,1,...,n 1. 5) The previous orthogonality conditions correspond to 4) and they could be obtained by the minimization of the integral n R ν=1 t τ ν ) s ν + dλt). The existence of such quadrature rules was proved by Chakalov [5], Popoviciu [38], Morelli and Verna [33], and existence and uniqueness by Ghizzetti and Ossicini [18].

4 4 Gradimir V. Milovanović and Marija P. Stanić The conditions 5) define a sequence of polynomials {π n,σ } n N0, π n,σ t) = n ν=1 t τ ν n,σ) ), τ n,σ) 1 < τ n,σ) < < τ n,σ) n, such that R π k,σ t) n ν=1 t τ ν n,σ) ) s ν +1 dλt) = 0, k = 0,1,...,n 1. These polynomials π k,σ are called σ-orthogonal polynomials, and they correspond to the sequence σ = s 1,s,...). Specially, for s = s 1 = s =, the σ orthogonal polynomials reduce to the s orthogonal polynomials. At the end of this section we mention a general problem investigated by Stancu [43, 44, 46]. Namely, let η 1,...,η m η 1 < < η m ) be given fixed or prescribed) nodes, with multiplicities l 1,...,l m, respectively, and τ 1,...,τ n τ 1 < < τ n ) be free nodes, with given multiplicities m 1,...,m n, respectively. Stancu considered interpolatory quadrature formulae of a general form I f ) = R f t)dλt) = n m ν 1 ν=1 i=0 A i,ν f i) τ ν ) + m ν=1 l ν 1 B i,ν f i) η ν ), 6) i=0 with an algebraic degree of exactness at least M + L 1, where M = n ν=1 m ν and L = m ν=1 l ν. Using free and fixed nodes we can introduce two polynomials Q M t): = n ν=1 t τ ν ) m ν and q L t): = m ν=1 t η ν ) l ν. Choosing the free nodes to increase the degree of exactness leads to the so called Gaussian type of quadratures. If the free or Gaussian) nodes τ 1,...,τ n are such that the quadrature rule 6) is exact for each f P M+L+n 1, then we call it the Gauss Stancu formula see [6]). Stancu [45] proved that τ 1,...,τ n are the Gaussian nodes if and only if t k Q M t)q L t)dλt) = 0, k = 0,1,...,n 1. 7) R Under some restrictions of node polynomials q L t) and Q M t) on the support interval of the measure dλt) we can give sufficient conditions for the existence of Gaussian nodes cf. Stancu [45] and [0]). For example, if the multiplicities of the Gaussian nodes are odd, e.g., m ν = s ν +1, ν = 1,...,n, and if the polynomial with fixed nodes q L t) does not change its sign in the support interval of the measure dλt), then, in this interval, there exist real distinct nodes τ ν, ν = 1,...,n. This condition for the polynomial q L t) means that the multiplicities of the internal fixed nodes must be even. Defining a new nonnegative) measure dˆλt) := q L t) dλt),

5 Quadrature rules with multiple nodes 5 the orthogonality conditions 7) can be expressed in a simpler form t k Q M t)dˆλt) = 0, k = 0,1,...,n 1. R This means that the general quadrature problem 6), under these conditions, can be reduced to a problem with only Gaussian nodes, but with respect to another modified measure. Computational methods for this purpose are based on Christoffel s theorem and described in details in [16] see also [0]). For more details about the concept of power orthogonality and the corresponding quadrature with multiple nodes and the maximal algebraic degree of exactness and about the numerical methods for their construction we refer readers to the article [5] and the references therein, as well as to a nice book by Shi [41]. The next natural generalization represents quadrature rules with multiple nodes and the maximal degree of exactness in some functional spaces which are different from the space of algebraic polynomials. A generalized Gaussian problem For a finite interval [a,b] and a positive integer N, Ghizzeti and Osicini in [19] considered quadrature rules for the computation of an integral of the type b a f x)wx)dx, where the weight function w and function f satisfy the following conditions: wx) L[a,b], f x) AC N 1 [a,b] AC m [a,b], m N 0, is the class of functions f whose m-th derivative is absolutely continuous in [a,b]). For a fixed number m 1 of points a x 1 < x < < x m b and a fixed linear differential operator E of order N of the form E = N k=0 a k x) dn k dx N k, where a 1 x) = 1, a k x) AC N k 1 [a,b], k = 1,,...,N 1, and a N x) La,b), they considered see [19, pp ]) a generalized Gaussian problem, i.e., they studied the quadrature rule b a f x)wx)dx = m ν=1 N 1 A j,ν, f j) x ν ) + R f ) j=0 such that E f ) = 0 implies R f ) = 0. According to [19, p. 8] quadrature rule which is exact for all algebraic polynomials of degree less than or equal ν 1 are relative to the differential operator E = dx dν ν, and quadrature rule which is exact for all trigonometric polynomials of degree less than or equal to ν are relative to the differential operator of order ν +1): E = dx d d ν k=1 + k ). dx

6 6 Gradimir V. Milovanović and Marija P. Stanić The considered whether there can exist a rule of the form b a f x)wx)dx = m ν=1 N p ν 1 A j,ν f j) x ν ) + R f ), 8) j=0 with fixed integers p ν, 0 p ν N 1, ν = 1,,...,m, such that at least one of the integers p ν is greater than or equal to 1, satisfying that E f ) = 0 implies R f ) = 0, too. The answer is given in the following theorem see [19, p. 45] for the proof). Theorem 1. For the given nodes x 1,x,...,x m, the linear differential operator E of order N and the nonnegative integers p 1, p,..., p m, 0 p ν N 1, ν = 1,,...,m, such that there exists ν {1,,...,m} for which p ν 1, consider the following homogenous boundary differential problem E f ) = 0, f j) x ν ) = 0, j = 0,1,...,N p ν 1, ν = 1,,...,m. 9) If this problem has no non trivial solutions whence N mn m ν=1 p ν) it is possible to write a quadrature rule of the type 8) with mn m ν=1 p ν N parameters chosen arbitrarily. If, on the other hand, the problem 9) has q linearly independent solutions U r, r = 1,,...,q, with N mn + m ν=1 p ν q p ν for all ν = 1,,...,m, then the formula 8) may apply only if the following q conditions b a U r x)wx)dx = 0, r = 1,,...,q, are satisfied; if so, mn m ν=1 p ν N + q parameters in the formula 8) can be chosen arbitrary. Since the quadrature rules with multiple nodes and the maximal algebraic degree of exactness are widespread in literature, in what follows we restrict our attention to the quadrature rules with multiple nodes and the maximal trigonometric degree of exactness. 3 Quadrature rules with multiple nodes and the maximal trigonometric degree of exactness For a nonnegative integer n and for γ {0,1/} by Tn γ we denote the linear span of the set {cosk + γ)x,sink + γ)x : k = 0,1,...,n}. It is obvious that Tn 0 = T n is the linear space of all trigonometric polynomials of degree less than or equal to n, is the linear space of all trigonometric polynomials of semi integer degree less than or equal to n + 1/ and dimtn) γ = n + γ) + 1. By T γ = n N 0 Tn γ we denote the set of all trigonometric polynomials for γ = 0) and the set of all trigonometric polynomials of semi integer degree for γ = 1/). For γ = 0 we simple write T instead of T 0. Finally, by T n we denote the linear space T n span{sinnx} or T n span{cos nx}. T 1/ n

7 Quadrature rules with multiple nodes 7 We use the following notation: γ = 1 γ, γ {0, 1/}. In what follows we simple say that trigonometric polynomial has degree k + γ, k N 0, γ {0,1/}, which always means that for γ = 0 it is trigonometric polynomial of precise degree k, while for γ = 1/ it is trigonometric polynomial of precise semi integer degree k + 1/. Obviously, every trigonometric polynomial of degree n + γ can be represented in the form T γ n x) = n k=0 c k cosk + γ)x + d k sink + γ)x), c n + d n 0, 10) where c k,d k R, k = 0,1,...,n. For γ = 0 we always set d 0 = 0. Coefficients c n and d n are called the leading coefficients. Let us suppose that w is a weight function, integrable and nonnegative on the interval [,π), vanishing there only on a set of a measure zero. The quadrature rules with simple nodes and the maximal trigonometric degree of exactness were considered by many authors. For a brief historical survey of available approaches for the construction of such quadrature rules we refer readers to the article [8] and the references therein. Following Turetzkii s approach see [49]), which is a simulation of the development of Gaussian quadrature rules for algebraic polynomials, quadrature rules with the maximal trigonometric degree of exactness with an odd number of nodes γ = 1/) and with an even number of nodes γ = 0) were considered in [8] see also [9]) and in [47], respectively. It is well known that in a case of quadrature rule with maximal algebraic degree of exactness the nodes are the zeros of the corresponding orthogonal algebraic polynomial. In a case of quadrature with an odd maximal trigonometric degree of exactness the nodes are the zeros of the corresponding orthogonal trigonometric polynomials, but for an even maximal trigonometric degree of exactness the nodes are not zeros of trigonometric polynomial, but zeros of the corresponding orthogonal trigonometric polynomial of semi integer degree. Similarly, when the quadrature rules with multiple nodes and the maximal algebraic degree of exactness are in question it is necessary to consider power orthogonality in the space of algebraic polynomials. But, if we want to consider quadrature rules with multiple nodes and the maximal trigonometric degree of exactness, we have to consider power orthogonality in the space of trigonometric polynomials or in the space of trigonometric polynomials of semi integer degree, which depends on number of nodes. In mentioned papers [8] and [47] the existence and the uniqueness of orthogonal trigonometric polynomials of integer or of semi integer degree were proved under the assumption that two of the coefficients in expansion 10) are given in advance it is usual to choose the leading coefficients in advance). Also, if we directly compute the zeros of orthogonal trigonometric polynomial of integer or semi integer degree), we can fix one of them in advance since for n+γ) zeros we have n + γ) 1 orthogonality conditions. Let n be positive integer, γ {0,1/}, s ν nonnegative integers for ν = γ, γ + 1,...,n, and σ = s γ,s γ+1,...,s n ). For the given weight function w we considered

8 8 Gradimir V. Milovanović and Marija P. Stanić a quadrature rule of the following type f x)wx)dx = n s ν ν= γ j=0 A j,ν f j) x ν ) + R n f ). 11) which have maximal trigonometric degree of exactness, i.e., for which R n f ) = 0 for all f T N1, where N 1 = n ν= γ s ν + 1) 1. In this case boundary differential problem 9) has the following form E f ) = 0, f j) x ν ) = 0, j = 0,1,...,s ν, ν = γ, γ + 1,...,n, 1) where E is the following differential operator of order N = N 1 + 1: ν= γ E = d dx N 1 k=1 ) d dx + k. The boundary problem 1) has the following n γ linear independent nontrivial solutions n U l x) = sin x x ) sν +1 ν cosl + γ)x, l = 0,1,...,n 1, V l x) = n ν= γ sin x x ν ) sν +1 sinl + γ)x, l = γ, γ + 1,...,n 1. According to Theorem 1, the nodes x γ,x γ+1,...,x n of the quadrature rule 11) satisfy conditions n ν= γ n ν= γ sin x x ν sin x x ν ) sν +1 cosl + γ)xwx)dx = 0, l = 0,1,...,n 1, ) sν +1 sinl + γ)xwx)dx = 0, l = γ, γ + 1,...,n 1, i.e., n ν= γ sin x x ) sν +1 ν t γ n 1 x)wx)dx = 0, for all tγ n 1 Tγ n 1. 13) Trigonometric polynomial T γ σ,n = n ν= γ sinx x ν)/ T γ n which satisfies orthogonality conditions 13) is called σ orthogonal trigonometric polynomial of degree n + γ with respect to weight function w on [,π). Therefore, the nodes of quadrature rule 11) with the maximal trigonometric degree of exactness are the zeros of the corresponding σ orthogonal trigonometric polynomial.

9 Quadrature rules with multiple nodes 9 Specially, for s γ = s γ+1 = = s n = s, σ orthogonality conditions 13) reduce to ) s+1 π n sin x x ν t γ n 1 x)wx)dx = 0, tγ n 1 Tγ n 1. 14) ν= γ Trigonometric polynomial T γ s,n = n ν= γ sinx x ν)/ T γ n which satisfies 14) is called s orthogonal trigonometric polynomial of degree n + γ with respect to w on [,π). For γ = 1/, s orthogonal trigonometric polynomials of semi integer degree were defined and considered in [7], while σ orthogonal trigonometric polynomials of semi integer degree were defined and their main properties were proved in [9]. For γ = 0, the both s and σ orthogonal trigonometric polynomials were defined and studied in [47]. It is obvious that s orthogonal trigonometric polynomials can be considered as a special case of σ orthogonal trigonometric polynomials. However, we first prove the existence and uniqueness of s orthogonal trigonometric polynomials because it can be done in a quite simple way, by using the well known facts about the best approximation given in the Remark 1 below see [10, p ]). The proofs of the existence and uniqueness of σ orthogonal trigonometric polynomials are more complicated. We prove them by using theory of implicitly defined orthogonality see [13, Sect. 5.3] for implicitly defined orthogonal algebraic polynomials). Remark 1. Let X be a Banach space and Y be a closed linear subspace of X. For each f X, inf g Y f g is the error of approximation of f by elements from Y. If there exists some g = g 0 Y for which that infimum is attained, then g 0 is called the best approximation to f from Y. For each finite dimensional subspace X n of X and each f X, there exists the best approximation to f from X n. In addition, if X is a strictly convex space, then each f X has at most one element of the best approximation in each closed linear subspace Y X. Analogously as it was proved in [8] and [47] for orthogonal trigonometric polynomials, one can prove the existence and uniqueness of s and σ orthogonal trigonometric polynomials fixing in advance two of their coefficients or one of their zeros. Before we prove the existence and uniqueness of s and σ orthogonal trigonometric polynomials, we prove that all of their zeros in [, π) are simple. Of course, it is enough to prove that for σ orthogonal trigonometric polynomials. Theorem. The σ orthogonal trigonometric polynomial T γ σ,n T γ n with respect to the weight function wx) on [,π) has in [,π) exactly n + γ) distinct simple zeros. Proof. It is easy to see that the trigonometric polynomial T γ σ,nx) must have at least one zero of odd multiplicity in [,π), because if we assume the contrary, then for n N we obtain that n ν= γ sin x x ν ) sν +1 cos0 + γ)xwx)dx = 0,

10 10 Gradimir V. Milovanović and Marija P. Stanić which is impossible, because the integrand does not change its sign on [,π). We know see [1] and []) that T γ σ,nx) must change its sign an even number of times for γ = 0 and an odd number of times for γ = 1/. First, we suppose that γ = 0 and that the number of zeros of odd multiplicities of T σ,n x) on [,π) is m, m < n. We denote these zeros by y 1,y,...,y m, and set tx) = m k=1 sinx y k)/). Since t T m, m < n, we have n ν=1 sin x x ν ) sν +1 tx)wx)dx = 0, which also gives a contradiction, since the integrand does not change its sign on [,π). Thus, T σ,n must have exactly n distinct simple zeros on [,π). Similarly, for γ = 1/ by y 1,y,...,y m+1, m < n, we denote the zeros of odd multiplicities of T 1/ σ,n x) on [,π) and set tx) = m+1 t T 1/ m, m < n, we have n ν=0 sin x x ν ) sν +1 x) tx)wx)dx = 0, k=1 sinx y k)/). Since which again gives a contradiction, since the integrand does not change its sign on [,π), which means that Tσ,n 1/ x) must have exactly n + 1 distinct simple zeros on [,π). 3.1 s orthogonal trigonometric polynomials Theorem 3. Trigonometric polynomial T γ s,nx) T γ n, γ {0,1/}, with given leading coefficients, which is s orthogonal on [, π) with respect to a given weight function w is determined uniquely. Proof. Let us set X = L s+ [,π], u = wx) 1/s+) c n,γ cosn + γ)x + d n,γ sinn + γ)x) L s+ [,π], and fix the following n γ linearly independent elements in L s+ [,π]: u j = wx) 1/s+) cos j + γ)x, j = 0,1,...,n 1, v k = wx) 1/s+) sink + γ)x, k = γ, γ + 1,...,n 1. Then Y = span{u j,v k : j = 0,1,...,n 1,k = γ, γ + 1,...,n 1} is a finite dimensional subspace of X and, according to Remark 1, for each element from X there exists the best approximation from Y, i.e., there exist n γ constants α j,γ, j = 0,1,...,n 1, and β k,γ, k = γ, γ + 1,...,n 1, such that the error

11 Quadrature rules with multiple nodes 11 n 1 n 1 ) u α k,γ u k + β k,γ v k = c n,γ cosn + γ)x + d n,γ sinn + γ)x k=0 k= γ n 1 n 1 s+ 1/s+) α k,γ cosk + γ)x + β k,γ sink + γ)x)) wx)dx), k=0 k= γ is minimal, i.e., for every n and for every choice of the leading coefficients c n,γ,d n,γ, c n,γ + d n,γ 0, there exists a trigonometric polynomial of degree n + γ such that T γ s,nx) =c n,γ cosn + γ)x + d n,γ sinn + γ)x n 1 n 1 α k,γ cosk + γ)x + k=0 k= γ T γ s,nx)) s+ wx)dx ) β k,γ sink + γ)x, is minimal. Since the space L s+ [,π] is strictly convex, according to Remark 1, the problem of the best approximation has the unique solution, i.e., the trigonometric polynomial Ts,n γ is unique. Therefore, for each of the following n γ functions F C k λ) = T γ s,n x) + λ cosk + γ)x ) s+ wx)dx, k = 0,1,...,n 1, Fk S λ) = T γ s,n x) + λ sink + γ)x ) s+ wx)dx, k = γ, γ + 1,...,n 1, its derivative must be equal to zero for λ = 0. Hence, T γ s,n x) ) s+1 cosk + γ)xwx)dx = 0, k = 0,1,...,n 1, T γ s,n x) ) s+1 sink + γ)xwx)dx = 0, k = γ, γ + 1,...,n 1, which means that the polynomial T γ s,nx) satisfies 14). 3. σ orthogonal trigonometric polynomials In order to prove the existence and uniqueness of σ orthogonal trigonometric polynomials we fix in advance the zero x γ = and consider the σ orthogonality conditions 13) as the system with unknown zeros. Theorem 4. Let w be the weight function on [,π) and let p be a nonnegative continuous function, vanishing only on a set of a measure zero. Then there exists a

12 1 Gradimir V. Milovanović and Marija P. Stanić trigonometric polynomial Tn γ, γ {0,1/}, of degree n + γ, orthogonal on [,π) to every trigonometric polynomial of degree less than or equal to n 1 + γ with respect to the weight function p Tn γ x) ) wx). Proof. By T γ n we denote the set of all trigonometric polynomials of degree n + γ which have n + γ) real distinct zeros = x γ < x γ+1 < < x n < π and denote S n γ = { x = x γ+1,x γ+,...,x n ) R n γ : < x γ+1 < x γ+ < < x n < π }. For a given function p and an arbitrary Q γ n T n, γ we introduce the inner product as follows f,g )Q γn = f x)gx) pq γ nx))wx)dx. It is obvious that there is one to one correspondence between the sets T γ n and S n γ. Indeed, for every element x = x γ+1,x γ+,...,x n ) S n γ and for Q γ nx) = cos x n sin x x ν 15) there exists a unique system of orthogonal trigonometric polynomials U γ k T γ k, k = 0,1,...,n, such that Un γ,cosk + γ)x ) Q γ = Un γ,sin j + γ)x ) n Q γ = 0, for all n k = 0,1,...,n 1, j = γ, γ + 1,...,n 1, and Un γ,un γ ) Q γ 0. n In such a way, we introduce a mapping F n : S n γ S n γ, defined in the following way: for any x = x γ+1,x γ+,...,x n ) S n γ we have F n x) = y, where y = y γ+1,y γ+,...,y n ) S n γ is such that,y γ+1,y γ+,...,y n are the zeros of the orthogonal trigonometric polynomial of degree n + γ with respect to the weight function pq γ nx))wx), where Q γ nx) is given by 15). It is easy to see that the function p cosx/) n sinx x ν)/) ) wx) is also an admissible weight function for arbitrary x = x γ+1,x γ+,...,x n ) S n γ \S n γ. Now, we prove that F n is continuous mapping on S n γ. Let x S n γ be an arbitrary point, {x m) }, m N, a convergent sequence of points from S n γ, which converges to x, y = F n x), and y m) = F n x m) ), m N. Let y S n γ be an arbitrary limit point of the sequence {y m) } when m. Thus,

13 Quadrature rules with multiple nodes 13 cos x cos x n p n p sin x ym) ν cos x n sin x ym) ν cos x n cosk + γ)x ) sin x xm) ν wx)dx = 0, k = 0,1,...,n 1, sin j + γ)x ) sin x xm) ν wx)dx = 0, j = γ, γ + 1,,...,n 1. Applying Lebesgue Theorem of dominant convergence see [3, p. 83]), when m + we obtain cos x cos x n p cos x n p sin x y ν n sin x y ν cos x n cosk + γ)x sin x x ν ) sin j + γ)x sin x x ν wx)dx = 0, k = 0,1,...,n 1, ) wx)dx = 0, j = γ, γ + 1,...,n 1, i.e., cosx/) n sinx y ν)/) is the trigonometric polynomial of degree n + γ which is orthogonal to all trigonometric polynomials from T γ n 1 with respect to the weight function p cosx/) n sinx x ν)/) ) wx) on [,π). Such trigonometric polynomial has n + γ) distinct simple zeros in [, π) see [8] and [47]). Therefore, y S n γ and y = F n x). Since y = F n x), because of the uniqueness we have y = y, i.e., the mapping F n is continuous on S n γ. Finally, we prove that the mapping F n has a fixed point. The mapping F n is continuous on the bounded, convex and closed set S n γ R n γ. Applying the Brouwer fixed point theorem see [34]) we conclude that there exists a fixed point of F n. Since F n x) S n γ for all x S n γ \S n γ, the fixed point of F n belongs to S n γ. If we denote the fixed point of F n by x = x γ+1,x γ+,...,x n ), then for T n x) = cosx/) n sinx x ν)/) we get T γ n x)cosk + γ)x p T γ n x) ) wx)dx = 0, k = 0,1,...,n 1, T γ n x)sin j + γ)x p T γ n x) ) wx)dx = 0, j = γ, γ + 1,...,n 1.

14 14 Gradimir V. Milovanović and Marija P. Stanić 3.3 Construction of quadrature rules with multiple nodes and the maximal trigonometric degree of exactness The construction of quadrature rule with multiple nodes and the maximal trigonometric degree of exactness is based on the application of some properties of the topological degree of mapping see [39]). For that purposes we use a certain modification of ideas of Bojanov [3], Shi [40], Shi and Xu [4] for algebraic σ orthogonal polynomials) and Dryanov [11]. Using a unique notation for γ {0, 1/} and omitting some details we present here results obtained in [9] and [47] for γ = 1/ and γ = 0, respectively. For a [0,1], n N, and Fx,a) = cos x ) as γ +1 n n sgn sin x x ν we consider the following problem sin x x ν ) as ν +1 t γ n 1 x)wx)dx, tγ n 1 Tγ n 1, Fx,a) = 0 for all t γ n 1 Tγ n 1, 16) with unknowns x γ+1,x γ+,...,x n. The σ orthogonality conditions 13) with x 1 = ) are equivalent to the problem 16) for a = 1, which means that the nodes of the quadrature rule 11) can be obtained as a solution of the problem 16) for a = 1. For a = 0 and a = 1 the problem 16) has the unique solution in the simplex S n γ. According to Theorem 4 we know that the problem 16) has solutions in the simplex S n γ for every a [0,1]. We will prove the uniqueness of the solution x S n γ of the problem 16) for all a 0,1). Let us denote and Wx,a,x) = n sin x x ν as ν + φ k x,a) = cos x ) as γ +1 Wx,a,x) sin x x wx) dx, 17) k for k = γ + 1, γ +,...,n. Applying the same arguments as in [11, Lemma 3.], the following auxiliary result can be easily proved. Lemma 1. There exists ε > 0 such that for every a [0,1] the solutions x of the problem 16) belong to the simplex S ε,γ = { y : ε y γ+1 + π,ε y γ+ y γ+1,...,ε y n y n 1,ε π y n }.

15 Quadrature rules with multiple nodes 15 Also, the following result holds see [9, Lemma.3] and [47, Lemma 4.4] for the proof). Lemma. The problem 16) and the following problem φ k x,a) = 0, k = γ + 1, γ +,...,n, 18) where φ k x,a) are given by 17), are equivalent in the simplex S n γ. It is important to emphasize that the problems 16) and 18) are equivalent in the simplex S ε,γ, for some ε > 0, as well as in the simplex S ε1,γ, for all 0 < ε 1 < ε, but they are not equivalent in S n γ. For the proof of the main result we need the following lemma for the proof see [9, Lemma.4]). Lemma 3. Let p ξ,η x) be a continuous function on [,π], which depends continuously on parameters ξ,η [c,d], i.e., if ξ m,η m ) approaches ξ 0,η 0 ) then the sequences p ξm,η m x) tends to p ξ0,η 0 x) for every fixed x. If the solution xξ,η) of the problem 18) with the weight function p ξ,η x)wx) is always unique for every ξ,η) [c,d], then the solution xξ,η) depends continuously on ξ,η) [c,d]. Now, we are ready to prove the main result. Theorem 5. The problem 18) has a unique solution in the simplex S n γ for all a [0,1]. Proof. We call the problem 18) as a;s γ+1,s γ+,...,s n ;w) problem and prove this theorem by mathematical induction on n. The uniqueness for n = 0 is trivial. As an induction hypothesis, we suppose that the a;s γ+1,s γ+,...,s n ;w) problem has a unique solution for every a [0,1] and for every weight function w, integrable and nonnegative on the interval [,π), vanishing there only on a set of a measure zero. For ξ,η) [,π] we define the weight functions p ξ,η x) = sin x ξ as n 1 + sin x η as n + wx). According to the induction hypothesis, the a;s γ+1,s γ+,...,s n ; p ξ,η ) problem has a unique solution x γ+1 ξ,η),x γ+ ξ,η),...,x n ξ,η)) for every a [0,1], such that < x γ+1 ξ,η) < < x n ξ,η) < π, and x ν ξ,η) for all ν = γ + 1, γ +,...,n depends continuously on ξ,η) [,π] see Lemma 3). We will prove that the solution of the a;s γ+1,s γ+,...,s n ;w) problem is unique for every a [0,1]. Let us denote x n 1 ξ,η) = ξ, x n ξ,η) = η, Wxξ,η),a,x) = n sin x x νξ,η) as ν +,

16 16 Gradimir V. Milovanović and Marija P. Stanić and φ k xξ,η),a) = cos x ) as γ +1 Wxξ,η),a,x) sin x x wx)dx, kξ,η) The induction hypothesis gives us that k = γ + 1,...,n. φ k xξ,η),a) = 0, k = γ + 1, γ +,...,n, 19) for ξ,η) D, where D = {ξ,η) : x n ξ,η) < ξ < η < π}. Let us now consider the following problem in D with unknown t = ξ,η): ϕ 1 t,a) = cos x ) as γ +1 Wxξ,η),a,x) sin x ξ wx)dx = 0, 0) ϕ t,a) = cos x ) as γ +1 Wxξ,η),a,x) sin x η wx)dx = 0. If ξ,η) D is a solution of the problem 0), then, according to 19), xξ,η) is a solution of the a;s γ+1,...,s n ;w) problem in the simplex S n γ. To the contrary, if x = x γ+1,x γ+,...,x n ) is a solution of the a;s γ+1,...,s n ;w) problem in the simplex S n γ, then x n 1,x n ) is a solution of the problem 0). Applying Lemma 1 we conclude that every solution of the problem 0) belongs to D ε = {ξ,η) : ε ξ x n ξ,η),ε η ξ,ε π η}, for some ε > 0. Let x be a solution of the problem a;s γ+1,s γ+,...,s n ;w) in S ε,γ. Then, differentiating ϕ 1 with respect to the x k, for all k = γ + 1, γ +,...,n,n, we have ϕ π 1 = as k + 1) cos x ) as γ +1 n x k n sgn sin x x ν sin x x ν as ν +1 ) n sin x x ν cos x x k wx) dx. ν k,ν n 1 Since n sin x x ν cos x x k T γ n 1, ν k,ν n 1 applying Lemma, we conclude that ϕ 1 x = 0, for all k = γ + 1,...,n,n. Further, applying elementary trigonometric transformations we get the following iden- k tity and for cos x y = cos x cos y + sin x y cos y sin y + cos x y sin y,

17 Quadrature rules with multiple nodes 17 I := ϕ 1 = as n π cos x ) as γ +1 n x n 1 sin x x ν as ν + we obtain where I 1 = cos x ) as γ +1 n I = cos x ν n 1 sin x x n 1 as n 1 cos x x n 1 wx) dx, cos x n 1 I = as n I 1 + sin x ) n 1 I, 1) ν n 1 ) as γ +1 n ν n 1 sin x x ν as ν + sin x x n 1 n 1 as cos x wx)dx, sin x x ν as ν + sin x x n 1 as n 1 sin x x n 1 wx) dx. Obviously, I 1 > 0 the integrand does not change its sign on [,π)) and I = 0 because of 0)). Since < x n 1 < π, from 1) we get ) ϕ1 sgn = 1. x n 1 Analogously, ϕ = 0, x k k = γ + 1, γ +,...,n 1, ϕ sgn x n ) = 1. Thus, at any solution t = ξ,η) of the problem 0) from D ε we have ϕ 1 ξ = n k= γ+1 k n 1 ϕ 1 x k x k ξ + ϕ 1 x n 1 = ϕ 1 x n 1, n 1 ϕ η = ϕ x k x k= γ+1 k η + ϕ = ϕ, x n x n ϕ 1 η = 0, ϕ ξ = 0. Hence, the determinant of the corresponding Jacobian Jt) is positive. We are going to prove that the problem 0) has a unique solution in D ε/. We prove it by using the topological degree of a mapping. Let us define the mapping ϕt,a) : D ε/ [0,1] R, by ϕt,a) = ϕ 1 t,a),ϕ t,a)), t R, a [0,1]. If xξ,η) is a solution of the a;s γ+1,...,s n ;w) problem in S ε,γ, then the solutions of the problem ϕt,a) = 0,0) in D ε/ belong to D ε. The problem ϕt,a) = 0,0) has

18 18 Gradimir V. Milovanović and Marija P. Stanić the unique solution in D ε/ for a = 0. It is obvious that ϕ,0) is differentiable on D ε/, the mapping ϕt,a) is continuous in D ε/ [0,1], and ϕt,a) 0,0) for all t D ε/ and a [0,1]. Then degϕ,a),d ε/,0,0)) = sgndetjt))) = 1, for all a [0,1], which means that degϕ,a),d ε/,0,0)) is a constant independent of a. Therefore, the problem ϕt,a) = 0,0) has the unique solution in D ε/ for all a [0,1]. Hence, the a;s γ+1,...,s n ;w) problem has a unique solution in S n γ which belongs to S ε,γ. Theorem 6. The solution x = xa) of the problem 18) depends continuously on a [0,1]. Proof. Let {a m }, a m [0,1], m N, be a convergent sequence, which converges to a [0,1]. Then for every a m, m N, there exists the unique solution xa m ) = x γ+1 a m ),...,x n a m )) of the system 18) with a = a m. The unique solution of system 18) for a = a is xa ) = x γ+1 a ),...,x n a )). Let x = x γ+1,...,x n ) be an arbitrary limit point of the sequence xa m ) when a m a. According to Theorem 5, for each m N we have cos x ) am s γ +1 Wxa m ),a m,x) sin x x wx)dx = 0, ka m ) When a m a the above equations lead to cos x ) a s γ +1 Wx,a,x) sin x wx)dx = 0, x k k = γ + 1, γ +,...,n. k = γ + 1, γ +,...,n. According to Theorem 5 we get that x = xa ), i.e., lim a m a xa m) = xa ). 4 Numerical method for construction of quadrature rules with multiple nodes and the maximal trigonometric degree of exactness Based on the previously given theoretical results we present the numerical method for construction of quadrature rules of the form 11), for which R n f ) = 0 for all f T N1, where N 1 = n ν= γ s ν +1) 1. The first step is a construction of nodes, and the second one is a construction of weights knowing nodes.

19 Quadrature rules with multiple nodes Construction of nodes We chose x γ = and obtain the nodes x γ+1,x γ+,,x n by solving system of nonlinear equations 18) for a = 1, applying Newton Kantorovič method. Obtained theoretical results suggest that for fixed n the system 18) can be solved progressively, for an increasing sequence of values for a, up to the target value a = 1. The solution for some a i) can be used as the initial iteration in Newton Kantorovič method for calculating the solution for a i+1), a i) < a i+1) 1. If for some chosen a i+1) Newton Kantorovič method does not converge, we decrease a i+1) such that it becomes convergent, which is always possible according to Theorem 6. Thus, we can set a i+1) = 1 in each step, and, if that iterative process is not convergent, we set a i+1) := a i+1) +a i) )/ until it becomes convergent. As the initial iteration for the first iterative process, for some a 1) > 0, we choose the zeros of the corresponding orthogonal trigonometric polynomial of degree n + γ, i.e., the solution of system 18) for a = 0. Let us introduce the following matrix notation Jacobian of φx), has the following entries φ i x j = 1 + as j ) x = [ ] T x γ+1 x γ+ x n, [ ] x m) = x m) T γ+1 xm) xm) γ+ n, m = 0,1,..., φx) = [ φ γ+1 x) φ γ+ x) φ n x) ] T. W = Wx) = [w i, j ] n γ) n γ) = cos x ) as γ +1 n n sin x x ν cos x x j wx)dx, ν i,ν j φ i = 1 + as i x i cos x ) as γ +1 n ν i sin x x i as i cos x x i wx)dx, [ φi+ γ x j+ γ sin x x ν sin x x ν ] n γ) n γ) as ν +1, n sgn sin x x ν i j, i, j = γ + 1, γ +,...,n, as ν + i = γ + 1, γ +,...,n. All of the above integrals can be computed by using a Gaussian type quadrature rule for trigonometric polynomials see [8] and [47]) )

20 0 Gradimir V. Milovanović and Marija P. Stanić f x)wx)dx = N ν= γ A ν f x ν ) + R N f ), with N n ν= γ s ν + n. The Newton Kantorovič method for calculating the zeros of the σ orthogonal trigonometric polynomial T γ σ,n is given as follows x m+1) = x m) W 1 x m) )φx m) ), m = 0,1,..., and for sufficiently good chosen initial approximation x 0), it has the quadratic convergence. 4. Construction of weights Knowing the nodes of quadrature rule 11), it is possible to calculate the corresponding weights. The weights can be calculating by using the Hermite trigonometric interpolation polynomial see [11, 1]). Since the construction of the Hermite interpolation trigonometric polynomial is more difficult than in the algebraic case, we use an adaptation on the method which was first given in [17] for construction of Gauss Turán quadrature rules for algebraic polynomials) and then generalized for Chakalov Popoviciu s type quadrature rules also for algebraic polynomials) in [30, 31]. For γ = 1/ and γ = 0 the corresponding method for construction of weights of quadrature rule 11) is presented in [9] and [47], respectively. Our method is based on the facts that quadrature rule 11) is of interpolatory type and that it is exact for all trigonometric polynomials of degree less than or equal to n ν= γ s ν + 1) 1 = n ν= γ s ν + n + γ) 1. Thus, the weights can be calculated requiring that quadrature rule 11) integrates exactly all trigonometric polynomials of degree less than or equal to n ν= γ s ν + n + γ) 1. Additionally, in the case γ = 0 i.e., γ = 1) we require that 11) integrates exactly cos n ν=1 s ν + n ) x, when n ν=1 s ν + 1)x ν = lπ for an odd integer l, or sin n ν=1 s ν + n ) x when n ν=1 s ν + 1)x ν = lπ for an even integer l, while if n ν=1 s ν + 1)x ν lπ, l Z, one can choose to require exactness for cos n ν=1 s ν + n ) x or sin n ν=1 s ν + n ) x arbitrary see [1,, 47]). In such a way we obtain a system of linear equations for the unknown weights. That system can be solved by decomposing into a set of n + γ) upper triangular systems. For γ = 0 we explain that decomposing method in the case when n ν=1 s ν + 1)x ν = lπ, l Z. Let us denote

21 Quadrature rules with multiple nodes 1 Ω ν x) = u k,ν x) = n i= γ i ν sin x x ) si +1 i, ν = γ, γ + 1,...,n, sin x x ν ) k Ω ν x), ν = γ, γ + 1,...,n, k = 0,1,...,s ν, and { u t k,ν x) = k,ν x)cos x x ν, k even, γ = 0, or k odd, γ = 1/, u k,ν x), k odd, γ = 0, or k even, γ = 1/, ) for all ν = γ, γ + 1,...,n, k = 0,1,...,s ν. It is easy to see that t k,ν is a trigonometric polynomial of degree less than or equal to n ν= γ s ν + n, which for γ = 0 has the leading term cos n ν=1 s ν + n ) x or sin n ν=1 s ν + n ) x, but not the both of them. Quadrature rule 11) is exact for all such trigonometric polynomials, i.e., R n t k,ν ) = 0 for all ν = γ, γ + 1,...,n, k = 0,1,...,s ν. By using the Leibniz differentiation formula it is easy to check that t j) k,ν x i) = 0, j = 0,1,...,s ν, for all i ν. Hence, for all k = 0,1,...,s ν we have s ν µ k,ν = t k,ν x)wx)dx = j=0 A j,ν t j) k,ν x ν), ν = γ, γ + 1,...,n. 3) In such a way, we obtain n + γ) independent systems for calculating the weights A j,ν, j = 0,1,...,s ν, ν = γ, γ +1,...,n. Here, we need to calculate the derivatives t j) k,ν x ν), k = 0,1,...,s ν, j = 0,1,...,s ν, for each ν = γ, γ + 1,...,n. For that purpose we use the following result see [9] for details). Lemma 4. For the trigonometric polynomials t k,ν, given by ) we have and for i > k t i) k,ν x ν) = 0, i < k; t k) k,ν x ν) = k! k Ω νx ν ), where t i) k,ν x ν) = v i) { v i) k,ν x ν), k even, γ = 0, or k odd, γ = 1/, u i) k,ν x ν), k odd, γ = 0, or k even, γ = 1/, k,ν x ν) = [i/] m=0 ) i 1) m ui m) m m k,ν x ν ), and the sequence u i) k,ν x ν), k N 0, i N 0, is the solution of the difference equation

22 Gradimir V. Milovanović and Marija P. Stanić f i) [i 1)/] ) i 1) m k,ν = i m 1) f m + 1 m+1 k 1,ν x ν ), ν = γ, γ +1,...,n, k N, i N 0, m=[i k)/] with the initial conditions f i) 0,ν = Ω i) ν x ν ). Finally, the remaining problem of calculating Ω i) ν x ν ), ν = γ, γ + 1,...,n, is solved in [9, Lemma 3.3]. Lemma 5. Let us denote Ω ν,k x) = sin x x ) si +1 i. i k i ν Then Ω ν,ν 1 x) = Ω ν,ν x) and Ω ν x) = Ω ν,n x). The sequence Ω i) ν,k,l x), l,i N 0, where Ω ν,k,l x) = sin x x ) l k+1 Ω ν,kx), k = γ, γ + 1,...,n 1, k ν 1, is the solution of the following difference equation f i) ν,k,l x) = i m=0 ) i 1 x m m sin xk+1 + mπ ) f i m) ν,k,l 1 x), l N 0, with the initial conditions f i) ν,k,0 = Ω i) ν,k, i N 0. The following equalities hold. Ω i) i) ν,k,s k+1 +1 x) = Ω ν,k+1,0 x), k ν 1, Ω ν i) x) = Ω i) i) ν,n x) = Ω ν,n 1,s n +1 x), ν n, Ω i) i) i) n x) = Ω n,n 1 x) = Ω n,n,s n 1 +1 x) According to Lemma 4, the systems 3) are the following upper triangular systems t 0,ν x ν ) t 0,ν x ν) t s ν ) 0,ν x ν ) A 0,ν µ 0,ν t 1,ν x ν) t s ν ) 1,ν x ν ) A 1,ν µ 1,ν. =, ν = γ, γ + 1,...,n t s ν ) s ν,ν x ν) A sν,ν µ sν,ν

23 Quadrature rules with multiple nodes 3 Remark. For the case γ = 0, if n ν=1 s ν + 1)x ν lπ, l Z, then we choose t k,ν, k = 0,1,...,s ν, ν = 1,,...,n, in one of the following ways or which provides that or t k,ν x) = u k,ν x)cos t k,ν x) = u k,ν x)sin x + n i=1 i ν s i + 1)x i + s ν x ν x + n s i + 1)x i + s ν x ν i=1 i ν { n ) } t k,ν T n ν=1 s ν +n cos span s ν + n x ν=1 { n ) } t k,ν T n ν=1 s ν +n sin span s ν + n x, ν=1 k = 0,1,...,s ν, ν = 1,,...,n, respectively., 4.3 Numerical example In this section we present one numerical example as illustration of the obtained theoretical results. Numerical results are obtained using our proposed progressive approach. For all computations we use MATHEMATICA and the software package OrthogonalPolynomials explained in [8] in double precision arithmetic. Example 1. We construct quadrature rule 11) for γ = 0, n = 3, σ = 3, 3, 3, 4, 4, 4), with respect to the weight function wx) = 1 + cosx, x [,π). For calculation of nodes we use proposed progressive approach with only two steps, i.e., two iterative processes: for a = 1/ with 9 iterations) and for a = 1 with 8 iterations). The nodes x ν, ν = 1,,...,6, are the following: , , , , , The corresponding weight coefficients A j,ν, j = 0,1,...,s ν, ν = 1,,...,6, are given in Table 1 numbers in parentheses denote decimal exponents).

24 4 Gradimir V. Milovanović and Marija P. Stanić j A j,1 A j, A j, ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) j A j,4 A j,5 A j, ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Table 1 Weight coefficients A j,ν, j = 0,1,...,s ν, ν = 1,,...,n, for wx) = 1 + cosx, x [,π), n = 3 and σ = 3,3,3,4,4,4). Acknowledgements The authors were supported in part by the Serbian Ministry of Education, Science and Technological Development grant number #174015). References 1. Cruz Barroso, R., Darius, L., Gonzáles Vera, P., Njåstad, O.: Quadrature rules for periodic integrands. Bi orthogonality and para orthogonality, Ann. Math. et Informancae ), Cruz Barroso, R., Gonzáles Vera, P., Njåstad, O.: On bi orthogonal systems of trigonometric functions and quadrature formulas for periodic integrands, Numer. Algor. 44 4) 007), Bojanov, B.D.: Oscillating polynomials of least L 1 -norm, In: G. Hämmerlin, ed., Numerical Integration, ISNM 57, Birkhäuser, Basel 198), Chakalov, L.: Über eine allgemeine Quadraturformel, C.R. Acad. Bulgar. Sci ), Chakalov, L.: General quadrature formulae of Gaussian type, Bulgar. Akad. Nauk Izv. Mat. Inst ), Bulgarian) [English transl. East J. Approx ), 61 76]. 6. Chakalov, L.: Formules générales de quadrature mécanique du type de Gauss, Colloq. Math ), Christoffel, E.B.: Über die Gaußische Quadratur und eine Verallgemeinerung derselben. J. Reine Angew. Math ), [Also in Ges. Math. Abhandlungen I, pp ] 8. Cvetković, A.S., Milovanović, G.V.: The Mathematica Package OrthogonalPolynomials, Facta Univ. Ser. Math. Inform ), Cvetković, A.S., Stanić, M.P.: Trigonometric orthogonal systems, In: Approximation and Computation In Honor of Gradimir V. Milovanović, Series: Springer Optimization and Its Applications, Vol. 4 W. Gautschi, G. Mastroianni, Th.M. Rassias eds.), pp , Springer Verlag, Berlin Heidelberg New York, DeVore, R.A., Lorentz, G.G.: Constructive Approximation, Springer Verlag, Berlin Heildeberg, 1993.

25 Quadrature rules with multiple nodes Dryanov, D.P.: Quadrature formulae with free nodes for periodic functions, Numer. Math ), Du, J., Han, H., Jin, G.: On trigonometric and paratrigonometric Hermite interpolation, J. Approx. Theory ), Engels, H.: Numerical Quadrature and Qubature, Academic Press, London, Gauss, C.F.: Methodus nova integralium valores per approximationem inveniendi, Commentationes Societatis Regiae Scientarium Göttingensis Recentiores ) [Also in Werke III, pp ] 15. Gautschi, W.: A survey of Gauss Christoffel quadrature formulae, in: P.L. Butzer, F. Fehér Eds.), E.B. Christoffel The Influence of his Work on Mathematics and the Physical Sciences, Birkhäuser, Basel, 1981, pp Gautschi, W.: Orthogonal polynomials: applications and computation, Acta Numerica 1996), Gautschi, W., Milovanović, G.V.: S orthogonality and construction of Gauss Turán-type quadrature formulae, J. Comput. Appl. Math ), Ghizzetti, A., Ossicini, A.: Sull esistenza e unicità delle formule di quadratura gaussiane, Rend. Mat. 6) ), Ghizzeti, A., Ossicini, A.: Quadrature Formulae, Academie Verlag, Berlin, Golub, G.H., Kautsky, J.: Calculation of Gauss quadratures with multiple free and fixed knots, Numer. Math ), Jacobi, C.G.J.: Ueber Gaußs neue Methode, die Werthe der Integrale näherungsweise zu finden. J. Reine Angew. Math ), Micchelli, C.A.: The fundamental theorem of algebra for monosplines with multiplicities, in: P. Butzer, J.P. Kahane, B.Sz. Nagy Eds.), Linear Operators and Approximation, ISNM vol. 0, Birkhäuser, Basel, 197, pp Milovanović, G.V.: Construction of s orthogonal polynomials and Turán quadrature formulae, in: G.V. Milovanović Ed.), Numerical Methods and Approximation Theory III Niš, 1987), Univ. Niš, Niš, 1988, pp Milovanović, G.V.: S orthogonality and generalized Turán quadratures: construction and applications, in: D.D. Stancu, Ch. Coman, W.W. Breckner, P. Blaga Eds.), Approximation and Optimization, vol. I Cluj-Napoca, 1996), Transilvania Press, Cluj-Napoca, Romania, 1997, pp Milovanović, G.V.: Quadrature with multiple nodes, power orthogonality, and moment preserving spline approximation, Numerical analysis 000, Vol. V, Quadrature and orthogonal polynomials W. Gautschi, F. Marcellan, and L. Reichel, eds.), J. Comput. Appl. Math ), Milovanović, G.V.: Numerical quadratures and orthogonal polynomials, Stud. Univ. Babeş- Bolyai Math ), Milovanović, G.V., Cvetković, A.S., Stanić, M.P.: Trigonometric orthogonal systems and quadrature formulae with maximal trigonometric degree of exactness, NMA 006, Lecture Notes in Comput. Sci ), Milovanović, G.V., Cvetković, A.S., Stanić, M.P.: Trigonometric orthogonal systems and quadrature formulae, Comput. Math. Appl ) 008), Milovanović, G.V., Cvetković, A.S., Stanić, M.P.: Quadrature formulae with multiple nodes and a maximal trigonometric degree of axactness, Numer. Math ), Milovanović, G.V., Spalević, M.M.: Construction of Chakalov Popoviciu s type quadrature formulae, Rend. Circ. Mat. Palermo, Serie II, Suppl ), Milovanović, G.V., Spalević, M.M., Cvetković, A.S.: Calculation of Gaussian type quadratures with multiple nodes, Math. Comput. Modelling ), Mirković, B.: Theory of Measures and Integrals, Naučna knjiga, Beograd, 1990 in Serbian). 33. Morelli, A., Verna, I.: Formula di quadratura in cui compaiono i valori della funzione e delle derivate con ordine massimo variabile da nodo a nodo, Rend. Circ. Mat. Palermo ) ), Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables, In: Classics in Applied Mathematics, vol. 30. SIAM, Philadelphia, 000.