Nonstandard Gaussian quadrature formulae based on operator values

Transcription

1 dv Comput Mat 2010) 32: DOI /s y Nonstandard Gaussian quadrature formulae based on operator values Gradimir V. Milovanović leksandar S. Cvetković Received: 23 July 2008 / ccepted: 11 January 2009 / Publised online: 24 February 2009 Springer Science + Business Media, LLC 2009 bstract In tis paper, we develop te teory of so-called nonstandard Gaussian quadrature formulae based on operator values for a general family of linear operators, acting of te space of algebraic polynomials, suc tat te degrees of polynomials are preserved. lso, we propose a stable numerical algoritm for constructing suc quadrature formulae. In particular, for some special classes of linear operators we obtain interesting explicit results connected wit teory of ortogonal polynomials. Keywords Gaussian quadrature Interval quadrature Linear operator Zeros Weigt Measure Degree of exactness Ortogonal polynomial Linear functional Matematics Subject Classifications 2000) C45 33D45 65D30 65D32 Communicated by Lotar Reicel. Te autors were supported in parts by te Serbian Ministry of Science and Tecnological Development Project #144004G Ortogonal Systems and pplications ). G. V. Milovanović B) Faculty of Computer Sciences, Megatrend University, Bulevar umetnosti 29, Novi Beograd, Serbia gvm@megatrend.edu.rs. S. Cvetković Faculty of Sciences and Matematics, University of Niš, P.O. Box 224, Niš, Serbia aleksandarcvetkovic@pmf.ni.ac.yu

2 432 G.V. Milovanović,.S. Cvetković 1 Introduction and preliminaries Let dμ be a finite positive Borel measure on te real line suc tat its support suppdμ) is an infinite set, and all its moments μ k = R xk dμx), k = 0, 1,..., exist and are finite. Wit P we denote te set of all algebraic polynomials and wit P n its subset formed by all polynomials of degree at most n N 0 ). Te n-point quadrature formula f x)dμx) = w k f x k ) + R n f ), 1.1) R k=1 wic is exact on te set P 2n 1, i.e., R n P 2n 1 ) = 0, is known as te Gauss quadrature formula cf. Gautsci [19, p. 29]). His famous metod of approximate integration, Gauss [17] discovered for te Legendre measure dμt) = dt on [ 1, 1] in 1814, and e obtained numerical values of quadrature parameters, te nodes x k and te weigts w k, k = 1,...,n, by solving nonlinear systems of equations for n 7. Computationally, today tere are very stable metods for generating Gaussian rules. Te most popular of tem is one due to Golub and Welsc [22]. Teir metod is based on determining te eigenvalues and te first components of te eigenvectors of a symmetric tridiagonal Jacobi matrix J n dμ), wit elements formed from te coefficients in te tree-term recurrence relation for te monic polynomials {π n dμ; )} + n=0 ortogonal wit respect to te inner product f, g) = f, g) dμ = f x)gx) dμx) f, g L 2 dμ)). 1.2) R Namely, te nodes x k in 1.1) are te eigenvalues of te Jacobi matrix J n dμ) i.e., zeros of π n dμ; )) and te weigts w k are given by w k = μ 0 v 2 k,1,were v k,1 is te first component in te corresponding normalized) eigenvector v k =[v k,1 v k,2... v k,n ] T ), vk Tv k = 1. Te Gaussian quadrature formulae were generalized in several ways. Te first idea of numerical integration involving multiple nodes appeared in te middle of te last century Cakalov [10 12], Turán [49], Popoviciu [43], Gizzetti and Ossicini [20, 21], etc.). survey on quadratures wit multiple nodes of te form 2s k f x)dμx) w k,i f i) x k ) R k=1 i=0 was recently publised by Milovanović [28]. Furter extensions dealing wit quadratures wit multiple nodes for ET Extended Tscebyceff) systems are given by Karlin and Pinkus [23], Barrow [2], Bojanov, Braess, and Dyn [6], Bojanov [5], etc. Recently, a metod for te construction of te generalized Gaussian quadrature rules for Müntz polynomials on 0, 1) is given in [32]. Te mentioned quadrature rules use te information on te integrand only at some selected points x k, k = 1,...,n te values of te function f and its

3 Nonstandard Gaussian quadrature 433 derivatives in te cases of rules wit multiple nodes). Suc quadratures will be called te standard quadrature formulae. However, in many cases in pysics and tecnics it is not possible to measure te exact value of te function f at points x k, so tat a standard quadrature cannot be applied. On te oter side, some oter information on f can be available, as 1 te averages 1 f x) dx 2 k I k of tis function over some non-overlapping subintervals I k,witlengtofi k equals 2 k, and teir union wic is a proper subset of suppdμ); 2 a fixed linear combination of te function values, e.g. afx ) + bfx) + cfx + ) at some points x k,werea, b, c are constants and is sufficiently small positive number, etc. Tus, if te information data { f x k )} n k=1 in te standard quadrature 1.1) is replaced by { k f )x k )} n k=1,were is an extension of some linear operator : P P, 0, we get a non-standard quadrature formula f x)dμx) = w k k f )x k ) + R n f ). 1.3) R k=1 Notice tat we use te same notation for te linear operator defined on te space of all algebraic polynomials and for its extension to te certain class of integrable functions X f X ). s a typical example for suc operators is te average Steklov) operator mentioned before in 1, i.e., p)x) = 1 x+ px) dx, > 0, p P. 1.4) 2 x Te first idea on so-called interval quadratures, wic are an example of nonstandard quadrature rules, appeared a few decades ago. In 1976 Omladičetal. [40] considered quadratures wit te average operator 1.4) see also Pitnauer and Reimer [41]). Some furter investigations were given by Kuz mina [25], Saripov [45], Babenko [1], and Motornyi [38]. Let 1,..., n be nonnegative numbers suc tat a < x 1 1 x < x 2 2 x < < x n n x n + n < b, 1.5) and let wx) be a given weigt function on [a, b]. Using te previous inequalities it is obvious tat we ave n )<b a.

4 434 G.V. Milovanović,.S. Cvetković Recently, Bojanov and Petrov [7] proved tat te Gaussian interval quadrature rule of te maximal algebraic degree of exactness 2n 1 exists, i.e., b a w k f x)wx) dx = 2 k k=1 xk + k x k k f x)wx) dx + R n f ), 1.6) were R n f ) = 0 for eac f P 2n 1. Under conditions k =, 1 k n,tey also proved te uniqueness of 1.6). Moreover, in [8] Bojanov and Petrov proved te uniqueness of 1.6) for te Legendre weigt wx) = 1) forany setoflengts k 0, k = 1,...,n, satisfying te condition 1.5). Te question of te existence for bounded a, b is proved in [7] in muc broader context for a given Cebysev system of functions. Recently in [29], using properties of te topological degree of non-linear mappings see [42, 44]), it was proved tat Gaussian interval quadrature formula is unique for te Jacobi weigt function wx) = 1 x) α 1 + x) β, α, β> 1,on[ 1, 1] and an algoritm for numerical construction was proposed. For te special case of te Cebysev weigt of te first kind and te special set of lengts an analytic solution can be given [29]. Interval quadrature rules of Gauss Radau and Gauss Lobatto type wit respect to te Jacobi weigt functions are considered in [33]. Recently, Bojanov and Petrov [9] proved te existence and uniqueness of te weigted Gaussian interval quadrature formula for a given system of continuously differentiable functions, wic constitute an ET system of order two on [a, b]. Te cases wit interval quadratures on unbounded intervals wit te classical generalized Laguerre and Hermite weigts ave been recently investigated by Milovanović and Cvetkovićin[30] and[34]. noter approac in quadrature formulae of Gaussian type for intervals of te same lengt wit te average operator 1.4) appeared in 1992 in Omladič s paper [39]. Te middle points of te intervals are zeros of some kind of ortogonal polynomials. More precisely, Omladič proved tat te nodes x k, k = 1,...,n, of is quadratures are zeros of te average Legendre polynomials p n x) p nx), wic satisfy te tree-term recurrence relation p n+1 x) = xp n x) n2 1 n 2 2) p 4n 2 n 1 x), n 1. 1 In tis paper we follow tis idea and develop te teory and numerical construction of nonstandard quadratures of Gaussian type for a general family of linear operators, acting of te space of algebraic polynomials, suc tat te degrees of polynomials are preserved. In particular, we consider some special linear operators, for wic we can get some interesting explicit results connected wit teory of ortogonal polynomials. Te paper is organized as follows. In te next section te formulation and te proof of te main result are given. Section 3 contain furter refinement of te teory, developed for some special classes of operators. Finally, using te

5 Nonstandard Gaussian quadrature 435 results presented in Section 3, Section4 resolves te problem of construction of tis kind of quadrature rules. 2 Nonstandard Gaussian quadrature formulae Let H = H δ be any rigt δ-neigborood of te number zero, i.e., H δ =[0,δ), δ>0. We consider families of linear operators, H, actingonte space of all algebraic polynomials P, suc tat te degrees of polynomials are preserved, i.e., and deg p) = degp), 2.1) lim 0 + p)x) = px), x C, 2.2) for any p P and eac H. Concerning degree preserving property for te convenience we define deg0) = 1, so tat degree preserving property also means tat te zero polynomial is te image only of te zero polynomial. For a given family of linear operators, H, we consider te nonstandard interpolatory quadrature of Gaussian type f x)dμx) = w k f )x k ) + R n f ), 2.3) R k=1 wic is exact for eac polynomial of degree at most 2n 1, i.e., R n P 2n 1 ) = 0. Our main result can be stated in te following form: Teorem 2.1 Let, H, be a family of linear operators satisfying te conditions 2.1) and 2.2) and dμ be a finite positive Borel measure on te real line wit its support suppdμ) R. Foranyn N tere exists ε>0, suc tat for every H ε =[0,ε) tere exists te unique interpolatory quadrature formula 2.3) of Gaussian type, wit nodes x k Cosuppμ)) and positive weigts w k > 0,k= 1,...,n. We are going to prove existence and uniqueness property of te nonstandard Gaussian quadrature formula 2.3) for a general family, H, satisfying te previous conditions. For some special classes of operators, suc as p)x) = m k= m p)x) = 1 x+ pt) dt, 2.4) 2 x a k px + k) or p)x) = m 1 k= m a k p x + k + 1 ) ), 2 2.5)

6 436 G.V. Milovanović,.S. Cvetković and p)x) = m b k k D k px), 2.6) k! we give more properties of te Gaussian quadrature formula. In te previous formulae we assume tat m is a fixed natural number and D k = d k /dx k, k N Some auxiliary results ccording to te fact tat we are working wit families of linear operators, we can represent te action of applied to any given p P if we know te values of tis operator to monomials x k, k N 0. Suppose tat we ave x n) t) = Ten it follows from 2.2)tat αk n )tk, n N 0. lim 0 αn + k ) = δ n,k, 0 k n, were δ n,k is te Kronecker s delta. To simplify te notation, we introduce te convention αk n ) = 0 for k > n. In tis way, we can state tis continuity property as follows: Lemma 2.1 Te family of linear operators, H, is continuous, i.e., lim 0 + p)t) = pt), p P, t C, if and only if te functions αk n ), k, n N, definedby x n )t) = k N 0 α n k )tk, n N 0, are continuous at zero, wit te property lim 0 + αn k ) = δ n,k, n, k N 0. Note tat tis continuity property in a certain sense can be formally written in te form lim 0 + = I,were,asusual,I is te identity operator. We will use tis simple notation to denote te continuity property. Lemma 2.2 degree preserving linear operator : P P is bijective. Proof Suppose we ave two polynomials p 1 and p 2 suc tat p 1 = p 2. ccording to te linearity p 1 p 2 ) = 0 and te degree preserving property degp 1 p 2 ) = 1, i.e., p 1 p 2 = 0, we conclude tat is injective.

7 Nonstandard Gaussian quadrature 437 If we suppose tat is not surjective, ten tere exists some polynomial q P for example, qt) := q k t k), suc tat it is not an image by of any p P. But degree preservation means tat in formulae x n )t) = k N 0 α n k t k, n N 0, we ave α n n = 0, n N 0. Suppose degq) = N. Ten we can solve te triangular system of equations x n )t) = k N 0 α n k t k, n = 0, 1,...,N, for te values t k, k = 0, 1,...,N, so tat we ave k t k = βn k x n )t), n=0 k = 0, 1,...,N. Ten, te polynomial N q k k n=0 β k n xn is mapped by into q, wic is a contradiction. Tis lemma sows tat we can treat our family of linear operators as a family of isomorpisms. Since every operator in te family is bijective, it as te inverse operator ) 1, wic is also linear cf. [26, p. 9]). Te following result is related to te inverse family of operators ) 1, H. Lemma 2.3 Let, H, be a given family of isomorpisms acting on te space of all algebraic polynomials, suc tat any operator preserves te degree of a polynomial and tat lim 0 + = I. Ten, te family of inverse operators ) 1, H, satisfies te same properties. Proof Suppose tat for some H, te operator ) 1 does not preserve te degree of polynomials, i.e., tere exists some p P, witdegp) = n, suc tat deg ) 1 p) = n. Ten, deg ) 1 p) = n implies deg ) 1 p)) = n, wic means tat does not preserve degree of te polynomial wic is a contradiction. s in te proof of Lemma 2.2 we can solve te triangular system of equations x n )t) = k N 0 α n k )t k, n = 0, 1,...,N,

8 438 G.V. Milovanović,.S. Cvetković so tat we ave t k = k βn k ) x n) t), n=0 pplying te inverse operator ) 1, we get ) 1 t k) x) = k βn k )xn, n=0 k = 0, 1,...,N. k = 0, 1,...,N. Using Cramer s formulae, te functions βn k ), 0 n k, can be expressed as rational functions of αk n), wic denominator is given by N ν=0 αν ν ) = 0, H, according to te degree preserving property. Since αk n), k, n N 0, are continuous at = 0, te functions βn k ), 0 k n, are also continuous at = 0. It means, according to Lemma 2.1, tat te family ) 1, H, is continuous. We also adopt te definition βn k ) = 0, n > k. s a direct consequence, we ave te following result: Lemma 2.4 Te functions αk n) and βk n ) satisfy te following simple property αν n )βν k ) = δ n,k, k, n N 0. ν N 0 Proof pplying te inverse operator to x n) t) = we get x n = x k ν=k k N 0 α n k )t k, αν n )βν k ). To complete te proof we need only to read te terms wit given powers of x. Now, for a given family, H, and a positive measure dμ, wit suppμ) R, we define te following family of te linear functionals L p) = ) 1 p)x) dμx), p P. 2.7) It is obvious tat te functional L, for a given H, is linear. We are now able to define te following bilinear functional p, q = L pq), p, q P, 2.8) wit properties wic are summarized in te following lemma.

9 Nonstandard Gaussian quadrature 439 Lemma 2.5 Te bilinear functionals 2.8) satisfy te following properties p, q = q, p, αp + βr, q = α p, q + β r, q, p, q, r P. Proof Direct calculation. ccording to tese properties, te bilinear functional, can be understood, in a general case, as a formal or non-hermitian inner product on P cf. [3, 46]). In a general case, te linear functional L is not regular, i.e., te sequence of formal) ortogonal polynomials wit respect to, does not exist. But, under certain assumptions te functional L could be regular, and we ave te following lemma. Lemma 2.6 Let te family, H, satisfy te property ) 1 : P, R) P, R). Ten, for every n N tere exists an ε>0, suc tat every linear functional L, [0,ε), is positive definite on P 2n, R) and te sequence of ortogonal polynomials π k,k= 0, 1,...,n, exists wit respect to L, [0,ε). Proof s in te proofs of te previous lemmas, we adopt te notation ) 1 t k) x) = βν k )xν, k = 0, 1,...,n, ν N 0 were te functions βν k ), according to Lemma 2.1, ave te property lim 0 βk + ν ) = δ k,ν. For te moments of te linear functional L, we ave m ν ) = L x ν ) = ) 1 t ν) ν x) dμx) = βν k )m k, ν = 0, 1,...,2n, were m k, k = 0, 1,...,2n, are te moments of te measure μ, i.e., practically te moments of te linear functional L 0. ccording to te property ) 1 : P, R) P, R), we know tat all moments are real. Te moments m ν ), ν = 0, 1,...,2n, are continuous functions of at te point = 0, and terefore we ave lim m ν) = m ν, ν = 0, 1,...,2n. 0 + ccording to te Teorem about positive definiteness of te linear functionals see [13, p. 15]), we can conclude tat a linear functional is positive definite on P 2n, R), provided all moments are real and te corresponding Hankel determinants m 0 ) m 1 ) m ν 1 ) m 1 ) m 2 ) m ν ) Δ ν ) =, ν = 1,...,n + 1, 2.9). m ν 1 ) m ν ) m 2ν 2 )

10 440 G.V. Milovanović,.S. Cvetković are positive Δ 0 ) := 1). For = 0 all determinants Δ ν 0), ν = 1,...,n + 1, are positive, since te measure μ is positive and te corresponding linear functional L 0 is positive definite. Te determinants Δ ν ), ν = 1,...,n + 1, are continuous functions of at te point = 0, and terefore tere exist ε ν, ν = 1,...,n + 1, suc tat Δ ν ) >0 for [0,ε ν ), ν = 1,...,n + 1. We can identify te set [0,ε)in te following form n+1 [0,ε)= [0,ε k ), ε = min{ε 1,...,ε n+1 }. k=1 Terefore, te family of te linear functionals L, [0,ε), is positive definite on P 2n, R); ence, tere exists te sequence of ortogonal polynomials πk, k = 0, 1,...,n, wit respect to eac L, [0,ε). ccording to te proof of tis lemma it is obvious tat ε depends of n. Namely, te following implication n 1 > n 2 [0,εn 1 )) [0,εn 2 )), is an immediate consequence. In anoter words, we can expect tat ε is a nonincreasing function of n. It is interesting to pose a question weter tere exists te case in wic εn) =+, n N. Later, we prove tat suc families of operators exist, for example, one family of operators fulfilling tis property is p)x) = 2px + /2) px /2), p P. For te families of linear functionals for wic ) 1 : P, R) P, C), in te general case we cannot claim tat te linear functional is positive definite, but we can prove tat some linear functionals L are regular. Lemma 2.7 For every given n N 0 tere exists an ε>0 suc tat te linear functional L, [0,ε), is regular on te space P 2n, C), i.e., tere exists a sequence of polynomials π ν, ν = 0, 1,...,n, ortogonal wit respect to L, [0,ε). Proof Like in te proof of te previous lemma, we start wit te moments of te linear functional L, m ν ) = L t ν ) = ) 1 t ν) x) dμx), ν = 0, 1,...,2n, and ten we form te Hankel determinants Δ ν ), ν = 1, 2,...,n + 1, asin 2.9). Since te measure μ is positive, te functional L 0 is regular. Using a continuity argument as in te previous lemma, we prove tat tere exists an ε>0, suc tat Δ ν ) = 0, ν = 1, 2...,n + 1,for [0,ε). Since te linear functionals L are regular on P 2n, C), we conclude tat te corresponding sequence of ortogonal polynomials exists.

11 Nonstandard Gaussian quadrature 441 Teorem 2.2 Suppose two families of linear operators 1 and 2 are given suc tat 1 2 ) p = 0, p P2n 1. Ten two different linear functionals L1 and L 2,definedby2.7), wicare induced by tese operators in te case tey are regular) ave te same first n members of te ortogonal polynomial sequence. In anoter words if we denote sequence of ortogonal polynomials wit respect to inner product 2.7) for = 1 wit p1 k,k N 0 and for = 2 wit p2 k,k N 0,ten p 1 k = p2 k, k = 0, 1,...,n. Proof It is enoug to prove tat first 2n 1 moments are te same for two inner products. Hence, it is enoug to prove tat te values of te operators 1 ) 1 and 2 ) 1 on 1, x,...,x 2n 1 are te same. s in te proofs of previous lemmas, we ave 1 t k) x) = 2 t k) x) = k αν k xν, k = 0, 1,...,2n 1. ν=0 Using tis system of equations we can solve for x ν, ν = 0, 1,...,n, since it is triangular system of equations wit αk k = 0. Using te fact tat 1 and 2 are linear, we get ν ) ν ) x ν = 1 βk ν t k = 2 βk ν t k, ν = 0, 1,...,2n 1. It is obvious tat 1 ) 1 2 ) 1 )x ν = 0, ν = 0, 1,...,2n 1. Hence, first 2n 1 moments of te linear functionals L1 and L 2 are te same. Using a representation of ortogonal polynomials via moments see [13, p. 17]) μ 0 μ 1... μ k μ 1 μ 2 μ k+1 p k x) =., μ k 1 μ k μ 2k 1 1 x x k we conclude tat p 1 k = p2 k, k = 0, 1,...,n. Tis result enables us to consider te family 2.4) as a special case of te family 2.5), since we can always construct a quadrature rule to suc tat for eac p P 2n 1. 1 x+ px)dx = 2 x n 1 k= n+1 w k px + k),

12 442 G.V. Milovanović,.S. Cvetković It also gives us a macinery to treat operator families of te following form 1 p)x) = px + t)dμt). μ[, ]) gain, tis family can be reduced to te family 2.5) ontespacep 2n 1,ifwe apply a similar interpolation quadrature rule. ctually, it can be proved tat te families 2.5)and2.6) are equivalent, in te sense given in te following lemma. Lemma 2.8 Let 1 be te family of operators given by 2.5).Givenn N tere always exists family of operators 2,givenby2.6), suc tat 1 2 ) p = 0, p P2n. Let 1 be te family of operators given by 2.6). Givenn N tere always exists family of operators 2,givenby2.5), suc tat 1 2 ) p = 0, p P2n. Proof It is enoug to construct an operator 2, given by 2.6), wic satisfies 1 2 ) x k = 0, k = 0, 1,...,2n. Te previous reduces to te linear system of equations k k ) b ν ν t k ν = ν ν=0 = m 1 l= m 1 a l t + l) k m 1 a l l= m 1 ν=0 k k ) t k ν l) ν, ν k = 0, 1,...,2n, i.e., m 1 b ν = a l l ν, ν = 0, 1,...,2n. 2.10) l= m 1 For te proof of te second part of te statement it is enoug to note tat equation 2.10) as te unique solution for a l, l = m 1,...,m 1, for te case m 1 = n, since te matrix of te system is a regular Vandermonde matrix. It is well-known tat te construction of a Gaussian quadrature formula is connected wit an interpolation problem. In our case, for any set of real distinct numbers x k, k = 1,...,n, and any operator from our family, we are concerned wit te solution of te following interpolation problem: Find

13 Nonstandard Gaussian quadrature 443 te polynomial P of degree less tan 2n wic solves te following system of equations P)x k ) = f 0,k, [ P)x)] x=x k = f 1,k, k = 1,...,n, 2.11) were f m,k m = 0, 1; k = 1,...,n) are any two given sequences of numbers. Lemma 2.9 Te interpolation problem 2.11) as te unique solution P P 2n 1. Proof t first, for real distinct numbers x k, k = 1,...,n, we construct polynomials n Hx) Hx) = x x k ), M k x) = x x k )H x k ), k=1 S k x) = 1 2M k x k)x x k ))M k x)) 2, T k x) = x x k )M k x)) 2, U k x) = ) 1 S k )x), V k x) = ) 1 T k )x), for wic te following properties can be verified by direct calculations ) [ U k xν ) = S k x ν ) = δ k,ν, ) ] U k x) x=x ν = S k x ν) = 0, V k ) xν ) = T k x ν ) = 0, [ V k ) x) ] x=x ν = T k x ν) = δ k,ν. Hence, we can identify te polynomial P in te following form Px) = [ f 0,k U k x) + f 1,k V k x)], P P 2n 1. k=1 It remains to prove tat our interpolation problem as te unique solution. Here it is enoug to verify tat te corresponding omogenous problem as only te trivial solution in P 2n 1. Suppose tat te omogenous problem as a solution P wic is not trivial. Ten, according to te system of equations P)x k ) = 0, [ P)x)] x=x k = 0, k = 1,...,n, we conclude easily tat te polynomial P as n distinct double zeros at te points x k, k = 1,...,n, wic means tat te polynomial P is of degree at least 2n. However, te operator preserves degree of polynomials, ence, te polynomial P as degree at least 2n, wic is a contradiction. Tus, our interpolation problem as te unique solution. 2.2 Proof of te main result Now, we are ready to prove our main result in te following reformulated form: Teorem 2.3 For every positive measure μ, wit suppμ) R, andanygiven family of isomorpisms, H, wic preserves degree of te polynomial, is continuous and, as te property ) 1 : P 2n, R) P 2n, R), for every

14 444 G.V. Milovanović,.S. Cvetković n N tere exists an ε>0, suc tat te quadrature formula 2.3) exists uniquely for [0,ε), i.e., pdμ = w k p)x k ), p P 2n 1, k=1 wit nodes x k Cosuppμ)) and positive weigts w k,k= 1,...,n. Proof s we proved in Lemma 2.6, te linear functional L, defined by 2.7), is positive definite on P 2n, R) for [0,ε 1 ), i.e., tere exists a sequence of polynomials ortogonal wit respect to L. We can express te monic ortogonal polynomials πn in te following form see [13, p. 17], [48, p. 97]) π 0 x) = 1, π n x) = 1 Δ n ) m 0 ) m 1 ) m n ) m 1 ) m 2 ) m n+1 ). m n 1 ) m n ) m 2n 1 ) 1 x x n, n 1, were Δ n ) is defined by 2.9). From tis formula we conclude tat te coefficients of polynomials πn x) are continuous functions of at te point = 0. Since te zeros of polynomials are continuous functions of teir coefficients see [35, p. 177]), we conclude tat te zeros of te polynomial πn are continuous functions of at = 0. Since all zeros of te polynomial πn 0 are contained in te set Cosuppμ)) R see [47, p. 4], [27]), ten according to te mentioned continuity property tere exists an ε 2 > 0 suc tat for [0,ε 2 ) te zeros of πn are contained in Cosuppμ)). Tus, for any [0,ε),wereε = min{ε 1,ε 2 }, we ave tat all zeros of πn are contained in Cosuppμ)). Now, take n N and a corresponding ε suc tat te linear functional L is positive definite on P 2n, R) and tat all zeros of πn are contained in Cosuppμ)). Coose some polynomial P P 2n 1. ccording to Lemma 2.9, we ave uniquely Px) = [ f 0,k U k x) + f 1,k V k x)], 2.12) k=1 wit f 0,k = P)x k ), f 1,k =[ P)x)] x=x k, k = 1,...,n, were te polynomials U k and V k are constructed for te set of points x k, k = 1,...,n, wic are zeros of πn. Using te definition of V k we ave V k dμ = ) 1 ) T k x) dμx) = L 1 T k ) = πn ) x k ) L πn M ) k = 0, according to te ortogonality property, since M k is of degree n 1. Te previous equality is true for every k = 1,...,n.

15 Nonstandard Gaussian quadrature 445 If we integrate 2.12), we get [ Pdμ = f 0,k U k dμ + f 1,k k=1 ] V k dμ = w k P)x k ), k=1 wit an identification w k = U k dμ, k = 1,...,n. In oter words, we ave just constructed a quadrature rule wic is exact for eac P P 2n 1. Its nodes are zeros of te polynomial πn ortogonal wit respect to L and, as we know, tese zeros belong to Cosuppσ )). For te weigt coefficients we ave w k = U k dμ = L S k ) = L M k ) 2) 2M k x k) πn ) x k ) L πn M ) k = L M k ) 2) > 0, were we ave used te ortogonality property and positive definiteness of te linear functional L. Te uniqueness property of our quadrature formula is identified easily, since te monic ortogonal polynomial πn is determined uniquely for te positive definite linear functional L, ence, its zeros are too. Tis completes te existence and te uniqueness property of te quadrature formula 2.3). However, tere exists still problem of te construction of suc a quadrature formula. It can be very instructive if we are able to give an algoritm for suc a construction or if we are able to derive a procedure using wic we can find te family ) 1, H. In te next section we present te procedure wic can resolve te mentioned questions for certain special families of operators. We finis tis section wit an illustrative example. Teorem 2.4 Suppose we ave te family of operators p)t) = pt) + t dpt), p P. dt Ten tis is a family of continuous, degree preserving, isomorpism of P. Te inverse family can be represented in te following form x 1/ x t 1/ 1 pt) dt, x > 0, ) 1 p ) 0 x) = p0), x = 0, x) 1/ 0 x t) 1/ 1 pt) dt, x < 0.

16 446 G.V. Milovanović,.S. Cvetković Proof Since te family is degree preserving it is a family of isomorpisms. Trivially we ceck it is continuous. We find easily te values of our family on te polynomial natural basis, x k) t) = 1 + k)t k i.e., ) 1 t k) x) = xk 1 + k, k N 0. Wat is left to do is just to ceck tat given representation of ) 1 matces given sequence of values on te natural basis. For example, tis teorem combined wit results already presented guaranties te existence of te following quadrature rule xk px) dμx) = w k x 1/ 1 px) dx, p P, R + k=1 x k 1 were x 0 = 0, for every measure μ and small enoug, depending on μ and n, were we can obtain te nodes of te quadrature formula as te zeros of te n-t polynomial ortogonal wit respect to te linear functional L p) = p ) x) dμx) = px) + x dpx) ) dμx), p P. R + R + dx It is also interesting to note tat te sequence of moments can be given in te form L x k) = 1 + k)l 0 x k ) = 1 + k) x k dμx), k N 0. R + 3 Special families of linear functionals 3.1 Basic consideration In order to construct te quadrature formula 2.3) we need te zeros of te polynomial πn ortogonal wit respect to te functional L defined by 2.7). Hence, te first problem is ow to compute te values of te functional L. ccording to te fact tat L is linear, we need only te moments of te linear functional L, i.e., we sould know ow to compute ) 1 x k, k = 0, 1,...,2n. If we know te action of, i.e., t n) x) = αk n xk, n N 0, k N 0 ten we can calculate t n = k N 0 α n k ) 1 x k) t), n N 0, and we are able to calculate te moments of te inverse operator ) 1, because te system of linear equations is triangular wit elements on te main

17 Nonstandard Gaussian quadrature 447 diagonal wic are not zeros. Since ) 1 is a linear operator, we know its action on te wole P. For a special families of linear operators, we are able to give more precise results on te interpretation of ) 1. In te sequel, we use te moments of te operators and ) 1,denoted by η n, x) = t n) x) and μ k, t) = ) 1 x k) t) n, k N 0 ), respectively. Note tat te first moment of is a constant different from zero. Teorem 3.1 ssume tat te moments of te operator can be expressed in te form η n, x) n! = were C k,k N 0, are constants. Ten η k, x) μ n k, t) = k! n k)! x k Cn k k! n k)!, n N 0, 3.1) x + t)n, n N 0, 3.2) n! and te moments of te inverse operators ) 1 satisfy te property 3.1). Te moments of te operator ) 1 are determined uniquely by te moments of and vice-versa. Proof pplying te operator ) 1 to 3.1), we get Next we ave t n n! = C n k n k)! η k, x) μ n k, t) = k! n k)! = = ν=0 ν=0 μ k, t), n N ) k! μ n k, t) n k)! x ν ν! x ν ν! k=ν k ν=0 t n ν n ν)! μ n k, t) n k)! x ν Ck ν ν! k ν)! C k ν k ν)! x + t)n =. n! Starting wit te equality 3.3), we note tat te matrix of te system of equations is lower-triangular and as constant elements on diagonals, were by diagonal we mean elements wic ave a constant difference of indices. We prove tat te inverse matrix of suc a matrix as te same properties. Trivially it is a lower-triangular matrix. Denote elements of tis inverse matrix by a i, j, i, j N 0. We know tat a i, j = 0 provided i < j. Te elements on te

18 448 G.V. Milovanović,.S. Cvetković main diagonal of tis inverse matrix are constant, since a i,i = 1/C 0, i N 0. Suppose tat elements on te i j = 0, 1,...,k. Using te identity we ave tat i ν= j a j+k+1+ν, j+ν = δ j +k+1+ν, j+ν C 0 = δ j+k+1, j C 0 C i ν i ν)! a ν, j = δ i, j, i, j N 0, 1 C 0 1 C 0 k l=0 j +k+ν l= j+ν C j +k+1+ν l j + k ν l)! a l, j+ν C k+1 l k + 1 l)! b l, j,ν N 0, wic means tat a j+k+1+ν, j+ν, ν N 0, does not depend on ν, i.e., te elements of te inverse matrix on te diagonal i j = k + 1 are constant. To prove te rest of tis teorem, we note tat te previous system of equations is triangular. If te moments of te operator are given, we ave μ n, t) n! = 1 η 0, x) x + t) n n! k=1 η k, x) k! μ n k, t) n k)! ), n N 0, were we use te fact tat η 0, x) = 0, since is a degree preserving operator. ccording to te fact tat ) 1 ) 1 =, te moments of are given uniquely by te moments of ) 1. Now, we introduce two generating functions for te moments of te operators and ) 1 by f u, x) = η k, x) u k, k! k N 0 f ) 1u, t) = μ k, t) u k, 3.4) k! k N 0 respectively. Tese functions are defined at least at te point u = 0. lso formally, we can form te Caucy product of tese two series wic represent te generating functions f u, x) f ) 1u, t) = k N 0 u k k ν=0 = k N 0 ux + t)) k k! Regarding tis we ave te following result: η ν, x) μ k ν, t) ν! k ν)! = expux + t)). Teorem 3.2 Let D = {0} be te domain of te absolute convergence of te series representing te generating function f u, x) from 3.4). Ten, te series wic represents te generating function f ) 1u, t) as a domain of te

19 Nonstandard Gaussian quadrature 449 absolute convergence D ) 1 = {0} and te bot generating functions satisfy te following equality f u, x) f ) 1u, t) = expux + t)), 3.5) were u D D ) 1, and tis intersection does not equal to {0}. Proof Define te function expux + t)) g u, t) :=, f u, x) wic is quotient of two analytic functions on D. Terefore, g is a meromorpic function of u on D. Because of f 0, x) = η 0 x) = 0, tere exists some neigborood D ) 1 of te point u = 0, suc tat te function f u, x) = 0 for u D ) 1. If te function f, x) does not take zero on te wole D, ten we can take D ) 1 = D. Te previous means tat te function g is analytic on D ) 1, i.e., it as te series representing it in te neigborood of u = 0. Hence, it must be f u, x)g u, t) = expux + t)), u D ) 1, were all functions in te formula are analytic. If we expand tese functions in potential series, we get η k,x) u k g ν, t) u ν = x + t) k u k. k! ν! k! k N 0 ν N 0 k N 0 For every u D ) 1, te product of series can be calculated using te Caucy product for f and g. ccording to tis fact, for u D ) 1 we ave u k η ν,x) g k ν, t) = expux + t)). ν! k ν)! k N 0 ν N 0 Finally, if we multiply it by u j 1, j N 0, and ten integrate it over te circle {u u =r} D ) 1, using Caucy s teorem, we obtain j ν=0 η ν, x) ν! g j ν, t) j ν)! j x + t) =, j N 0. j! ccording to Teorem 3.1, te previous system of equations uniquely determines te moments of ) 1,sotat f ) 1 = g and te function f ) 1 is analytic on D ) Special cases of families 2.4), 2.5), and 2.6) Knowing te generating function of moments of te operator,wecan determine te corresponding generating function of moments of te operator ) 1. For te families given by 2.4), 2.5), and 2.6), we can prove tat tey satisfy te previous property 3.1).

20 450 G.V. Milovanović,.S. Cvetković Teorem 3.3 Te families of operators given by 2.4), 2.5),and2.6) satisfy te property 3.1). Proof Using a direct calculation, for te family given by 2.4) we ave η k, x) k! were = 1 2k! x+ x t k dt = 1 x + ) k+1 x ) k+1 = 2 k + 1)! C ν = 1 + 1)ν 2ν + 1) ν, ν N 0. For te first family given in 2.5), we ave k ν=0 x ν C k ν ν! k ν)!, η k, x) k! = m ν= m a ν k j=0 x j ν) k j k j! k j)! = x j Ck j j! k j)!, j=0 were C ν = m j= m a j j) ν, ν N 0. Similarly, we ave for te second family given in 2.5). Finally, for te family given by 2.6), we ave η k, x) k! = k b ν ν ν=0 ν! x k ν m k ν)! = ν=0 x ν ν! C k ν k ν)!, were C ν = b ν ν, ν N 0. In order to simplify te notation we introduce te following definition: Definition 3.1 For te families of operators defined by p = m k= m a k p +k) and p = we introduce te caracteristic polynomials as Qz) = m k= m respectively. For te family m 1 k= m a k z k+m and Qz) = p = m b k k D k p, k! a k p + k + 1 ) ), 2 m 1 k= m a k z k+m, 3.6)

21 Nonstandard Gaussian quadrature 451 we define te caracteristic polynomial as m b k Qz) = k! zk. 3.7) For te family given in 2.4) it is proven in [39] tat it is a bijective family acting on P. Te degree preserving and continuity properties of tis family are trivial. For families of te form 2.5) and2.6), we ave te following statement: Teorem Te family of operators,definedby2.5), is bijective family of continuous and degree preserving operators if and only if for te caracteristic polynomial 3.6) we ave Q1) = 1. 2 Te family of operators,definedby2.6), is bijective family of continuous and degree preserving operators if and only if for te caracteristic polynomial 3.7) we ave Q0) = 1. Proof 1 It is enoug to consider only te first family of operators in 2.5). If te family 2.5) is continuous ten 0 = I, but m p = 0 p = a k p +k0) = Q1)p, p P, k= m ence, Q1) = 1. Now, let Q1) = 1. Te family 2.5) is evidently linear. It is a degree preserving as well, since te leading coefficients in te polynomials p and p are te same. Te family is bijective according to Lemma 2.2. Finally, for te continuity we ave first 0 = I and, because of linearity, we know tat it is enoug to prove te continuity only for one basis of P, so tat t j) m m j j ) x) = a k x + k) j = x ν k) j ν ν k= m k= m j = x j j ) m a k k) j ν. ν ν=0 k= m Since j ) m lim a k k) j ν = δ j,ν, 0 + ν k= m we conclude tat te family is continuous. 2 similar proof can be given for te family of operators 2.6). a k ν=0

22 452 G.V. Milovanović,.S. Cvetković Teorem Te generating functions f and f ) 1 of te operator 2.5) are given by f u, x) = Q e u) e ux exp degq)u/2) and exput + degq)/2)) f ) 1u, t) =, Qe u ) respectively, were Q is defined in 3.6). 2 Te generating functions for te family 2.6) are given by f u, x) = Qu)e ux and eut f ) 1u, t) = Qu), were Q is defined by 3.7). 3 Te generating functions for te family 2.4) are given by ux sinu) f u, x) = e u and f ) 1u, t) = eut u sinu). Proof We will determine only te expressions for f u, x). Te expressions for f ) 1u, t) are obtained directly from 3.5). 1 For te first family of 2.5) we get f u, x) = = + u k k! m a j x + j) k = j= m m + ux) k a j k! j= m + m=0 m + a j j= m m=0 k ux) m m! ju) m m = e ux a j e ju m! j= m ju) k m k m)! = Q e u) e ux exp degq)u/2), were we ave used intensively Caucy series product teorem and Fubini teorem. 2 For te family given by 2.6), we ave f u, x) = = + minm,k) u k j=0 m b j u) j j=0 j! b j j + j! x k j k j)! ux) k = Qu)e ux. k! 3 Finally, for te family 2.4) we obtain f u, x) = 1 + u k+1 x + ) k+1 x ) k+1 ux sinu) = e. 2u k! k + 1 u

23 Nonstandard Gaussian quadrature 453 Remark 3.1 It may be interesting to give precisely te domain on wic equality 3.5) is valid. Since te functions f and exp are entire functions, te expression is valid on te neigborood of te number 0 on wic te function f ) 1 is analytic. For te family 2.5), te equality is valid on te following set { { u u < min log λν +iargλ ν ) + 2kπ) } } = u min, k N 0,ν=1,...,degQ) were λ ν, ν = 1,...,degQ), are zeros of te caracteristic polynomial Q, counting multiplicities. Similarly, for te family 2.6), te equality is valid on te set { { u u < min λν } } = u min, ν=1,...,degq) were, again λ ν, ν = 1,...,degQ), are zeros of te caracteristic polynomial Q, counting multiplicities. From tese expressions we can give te following estimate μ k, t) 1/k k, eu min for te rate of increasing of te moments μ k, t). 3.3 Representation of inverse families of operators Using every operator we can derive te family of te linear functionals x acting on te space of algebraic polynomials in te following way x p = p)x), p P. Te linearity of te functionals x is a direct consequence of te linearity of. lso, we can introduce te moments ηk, x, k N 0, of te functionals x, as well as μk, x = μ k,x) for ) 1. Te generating functions for te moments ηk, x and μt k, of te functionals x and ) 1, t we denote by f u) and x 1, respectively. Tere are te obvious connections f t ) f x u) = f u, x) and f t ) 1u) = f ) 1u, t). Definition 3.2 For te measure μ possibly complex), we say it represents a linear functional L : P C, provided L x k) = x k dμx), were Γ is a simple Jordan curve in te complex plane. Γ In a general case, according to Teorem of representation of te complex linear functionals see [13, p. 74]), we can claim tat every linear functional as an interpretation measure, wic is even supported on te subset of te

24 454 G.V. Milovanović,.S. Cvetković real line. We ave just mention, tat for positive definite linear functionals, if te representation measure is supported on a compact set of te real line, te representation measure is unique see [13, p.71],[26, p. 410]). If te supporting set is unbounded, te representing measure need not be unique see [13, p. 73]), but tere are some sufficient conditions for tis measure to be unique for example, see [3, 16]). Since, we are interested to interpret te family of te linear functionals ), t as a result we sould get a family of te representing measures μ t, wit te property ) x k t = x k dμt x). Γ However, wen a family of operators satisfies te property 3.1), it is possible to get te representation using only one measure. Teorem 3.6 If a family of linear operators satisfies te property 3.1) we ave te representation of ) t in te form ) x k t = x + t) k dμ x), 3.8) were Γ is a simple Jordan curve in te complex plane. Proof ccording to Teorem 3.1, for some sequence B k, k N 0, we ave μ n, t) n! = Γ B n k t k n k)! k!, n N 0. Now, we suppose tat te measure μ, supported on some curve Γ,astemoments Bk, k N 0. Suc a measure always exists according to [13, pp ]. Ten, obviously we ave x + t) n dμ t k 1 x) = n! x n k dμ x) k! n k)! Γ = n! B n k t k Γ n k)! k! = μ n,t), n N 0. direct consequence of te previous teorem is tat te linear functional L, defined in 2.7), can be represented as L p) = dμt) pt + x)dμ x), p P. Γ

25 Nonstandard Gaussian quadrature 455 Lemma 3.1 If te family of measures μ is a family of positive measures supported on te subset of te real line for [0,ε) R + 0,tenL is a positive definite for [0,ε). Proof ssume tat px) is non-negative for any x R. Ten, for a fixed t R so is pt + x) for x R. Since positive measures represent positive definite functionals, for given t R we ave qt) = pt + x)dμ x) >0, t R. Γ Since q is a non-negative polynomial and te measure μ is positive we ave also L p) >0, wic implies L is positive definite see [13, p. 13]). We are going to see in te next section tat suc situations actually appen. Even more we are going to see tat tere are cases in wic ε =+. Positive definiteness of L guaranties te existence of te quadrature rule 2.3) wit te real nodes and positive weigts for any [0,ε), altoug it migt appen nodes are not contained in te Cosuppμ)). ssuming te measure μ as a support wic is unbounded towards ±, te positive definiteness of L will produce a quadrature rule wit nodes inside of te convex ull of te supporting set. Teorem 3.6 actually means tat we ave ) x n t = 0 ) 1x + t) n, n N 0, were ) 0 operates on x. ctually, we can prove tat 3.1) is equivalent wit te previous property. Lemma 3.2 Te sequence of moments η n, t), n N 0, of a linear operator satisfies 3.1) if and only if for te associated linear functionals we ave t were te operators are acting on x. px) = 0 px + t), 3.9) Proof Due to linearity it is enoug to give te proof only for te natural basis. ssume 3.1) olds, ten putting t = 0, we get so tat t x n = n! 0 xn = C n, n N 0, t k 0 xn k k! n k)! = 0 x + t)n, n N 0. n!

26 456 G.V. Milovanović,.S. Cvetković If 3.9) olds, we can use te linearity of 0 1 n! t x n = 1 n! 0 x + t)n = to obtain t k 0 xn k k! n k)!, n N 0. Finally, coosing C n = 0 xn, n N 0, we get 3.1). We are going to illustrate tis fact for te operators given by 2.4), 2.5) and 2.6). In order to give te results for te mentioned operators we need te following auxiliary result. Lemma 3.3 Let te function f : R C be infinitely continuously-differentiable, i.e. f C R), wit all derivatives integrable on R, and assume its Fourier transform is given by 2πw. Ten polynomialsareintegrablewit respect to χ R x)wx)dx, and we ave f n) 0) = ix) n wx)dx, n N 0. R Proof Since f C it is te well-known tat, for any k N 0, tere exist a positive constants C k suc tat wx) C k 1 + x ) k, x R. Tis property guaranties tat all polynomials are integrable wit respect to te measure χ R x)wx)dx. Using te continuity of f, we ave te Fourier inversion formula f u) = 1 e iux 2πwx)dx, u R. 3.10) 2π R ccording to Lebesgue teorem of dominant convergence, we can differentiate 3.10), so tat we ave f n) 0) = ix) n wx)dx. For te function wic dominates x k wx), k N 0,wecantake R C k+2 x k 1 + x ) k+2, k N 0, wic is integrable on R. Tis lemma togeter wit Lemma 3.2, suggests tat we can searc for te representation of te inverse family t ) 1, in te following way: first take te

27 Nonstandard Gaussian quadrature 457 generating function f t ) 1 use Lemma 3.2 to recover at t = 0, find te Fourier transform of it and ten t ) 1 for all t. In te sequel we consider te representation of te inverse operators for te mentioned families of operators. In order to be able to present results in a simpler way we introduce te following definition. Definition 3.3 Wit λ ν, ν = 1,...,M, we denote distinct zeros of te caracteristic polynomial Q of a family of linear operators,andwitm ν, ν = 1,...,M, teir multiplicities, respectively, so tat were M ν=1 m ν = degq). Qx) = M x λ ν ) m ν, 3.11) ν= Representation of te inverse family for 2.6) Since we work only wit bijective operators, according to Teorem 3.4, we must ave Q0) = 1. We can obtain te moments of te linear functional ) t by expanding te moment generating function f u) t ) into a power series in te neigborood of u = 0. Since,Q is a polynomial, at first we can expand te expression 1/Q into te partial fraction decomposition, and ten expand every term into te power series at te point u = 0. ccording to 3.11) we ave te following partial fraction decomposition and te series expansion were 1 M Qu) = ν=1 m ν j=1 Qν j + u k = u λ ν ) j k! μ ν, j k, = 1) j k j ) k. λ j+k ν M m ν ν=1 j=1 Q j ν μν, j k,, 3.12) Symbol j ) k = Γj + k)/γ j ) is Pocammer s symbol. We can conclude tat it is enoug to give a representation of te linear functional L λ,m t,, wit a caracteristic polynomial ) λ z m Q := Q m λ z) =. λ Ten we obviously ave ) M m ν 1) j Qν j t = ν=1 j=1 λ j ν L λ ν, j t,. 3.13)

28 458 G.V. Milovanović,.S. Cvetković For a representation of te previous linear functional we ave te following result: Teorem 3.7 Let rλ) = sgnrλ)) and iλ) = sgniλ)) and let a family of linear operators, defined by 2.6), avetecaracteristicpolynomialdetermined by Q := Q m λ z). 1 For Rλ) = 0, we ave ) ) λrλ) m 1 t p = pt + yrλ))y m 1 e λyrλ)/ dy. Γm) R + 2 For Rλ) = 0, we ave ) ) iλiλ) m 1 t p = pt iyiλ))y m 1 e iλyiλ)/ dy. Γm) R + Proof ccording to Lemma 3.2, it is enoug to give a representation for ). 0 We are not going to use metod based on te Fourier transform, given in Lemma 3.3, since for m = 1, we clearly does not ave integrability of te function 1/Q 1 λ. Te moments of te functional ) can be obtained easily. Namely, we ave f 0 ) 1u) = 1 Q m λ u) = u/λ ) m = Γm + k) k!γm) k N 0 ) u k, λ so tat te moments are Γ m + k)/γ m))/λ) k, k N 0. We present te proof only in te case Rλ) > 0 and Iλ) > 0. For oter cases te proof is almost te same. Let te contour C in te complex y-plane be te union of te following arcs γ1 R γ3 R ={y 0 y R}, γ R 2 ={y 0 y R, argy) = argλ)}. ={y y =R, argλ) argy) 0}, Te function y G k y) = y k y m 1 e λy/, k N 0, is analytic, except for te singularity at y =. ccording to te Caucy residue teorem, we ave C G ky) dy = 0. For te integral over te arc γ3 R, we ave R G k y) dy = t m+k 1 e im+k) argλ) e λ t/ dt γ R 3 = 0 ) m+k λ R/ λ 0 t m+k 1 e t dt ) m+k Γm + k), λ as R +.Itissimpletoprovetat γ 0 as R +, wic implies 2 R γ = + 1 γ. Using te integral calculated over γ + 3 R as te value of te integral 3 over γ1 R, after multiplication wit te constant from te statement, we get exactly te moments we need.

29 Nonstandard Gaussian quadrature 459 Note tat te representation teorem recovers te well-known generalized Laguerre measure. lso, note tat in te case Q := Q m λ z) and λ>0, te representation measure is positive. We ave te following result: Teorem 3.8 Suppose all te roots of te caracteristic polynomial Q, for te family given by 2.6), are positive. Te linear functional ) t as a representation given by ) t p = px + t) dμ R x), > 0, p P, + were te measure μ is positive. Proof ccording to Teorem 3.7, we can always represent te measure μ from tis teorem as a linear combination of te generalized Laguerre measures. So, we searc for te measure μ in an absolutely continuous form wm 1,...,m M x)dx, were we assume te notation from Definition 3.3. Let M ν=1 m ν > 2. Ten, using Lemma 3.3, we ave i.e., w m 1,...,m M x) = 1 2π R iue iux Q m1,...,m M iu) du w m 1,...,m M x) + λ M w m 1,...,m M x) = λ M e ixu 2π R Q m1,...,m M 1iu) du = λ M w m 1,...,m M 1x). ccording to te fact tat all singularities are placed in te lower alf plane and Q m1,...,m M iu) is of order at least u 3 at infinity, we ave w m1,...,m M 0) = 1 2π R du Q m1,...,m M iu) = 0. Te previous means tat our function w m1,...,m M is a solution of te following differential equation w m 1,...,m M x) + λ M w m 1,...,m M x) = λ M w m 1,...,m M 1x), w m1,...,m M 0) = 0, for wic we can give te explicit solution in te form w m1,...,m M x) = λ M e λ Mx/ e λmt/ w m1,...,m M 1t) dt. 0 It is clear tat if w m1,...,m M 1x) >0, x > 0, andw m1,...,m M 1x) = 0, x 0, ten also w m1,...,m M as te same properties. Now, we can apply an inductive argument. x

30 460 G.V. Milovanović,.S. Cvetković Wat is left to prove is tat te inductive base is true. Tus, we need to prove te statement of teorem for M m ν 2. ν=1 If te previous sum is one we ave already proved it in te representation teorem for te caracteristic polynomial Q := Q 1 λ z). If te previous sum equals two we distinguis two cases. Te first case, wen m 1 = 2,isalsoproved using te representation teorem for Q := Q 2 λ z), and te second one for wic λ 1 <λ 2 and m 1 = m 2 = 1, M = 2. For tis case we calculate directly w 1,1 and we get w 1,1 x) = 1 λ 1 λ 2 e ixu 2π R λ 1 iu)λ 2 iu) du, i.e., 1 λ 1 λ 2 e λ 1 x/ e λ 2x/ ), x > 0, w 1,1 x) = λ 2 λ 1 0, x 0. Tus, we convince ourself tat inductive base is satisfied. Using Lemma 2.6, we can interpret tis result also in te following form: Teorem 3.9 ssume tat all zeros of te caracteristic polynomial Q are positive, ten for R + and any positive measure μ, te linear functional L p) = ) 1 p ) x) dμx), p P, is positive definite Representation of te inverse family for 2.5) For te family given by 2.5) we can use also te partial fraction decomposition to get 1 M Qz) = ν=1 m ν j=1 Q j ν z λ ν ) j. Using series expansions for te functions 1/e u λ ν ) j, j N 0, ν = 1,...,M, in te form 1 e u λ ν ) j = k N 0 u k k! μν, j k,,

31 Nonstandard Gaussian quadrature 461 we obtain 1 Qe u ) = u k k! k N 0 M m ν ν=1 j=1 Q j ν μν, j k,. Tus, our problem of te representation is reduced to a representation of te family of linear functionals aving te caracteristic polynomial of te form ) z λ m Q := Q m λ z) =. 1 λ However, we must distinguis te cases λ =1 and λ = 1. In general, since Q1) = 1, we know tat λ ν = 1, ν = 1,...,degQ).) Teorem 3.10 For λ = 1, te linear functional ) 1, t wit te caracteristic polynomial Q := Q m λ z), as te following representation t ) 1 p = 1 λ) m + j 1 ) m λ j pt j + m/2)), λ < 1, j j N ) m λ j N 0 m + j 1 j ) pt + j + m/2)) λ j, λ > 1. Proof If we expand f t ) at te point u = 0,for λ > 1 we ave f u) t ) = 1 λ)m + u k t + m/2)k k! ) 1 m + m λ j j=0 ) e ju λ) j = = 1 1 ) m + u k + u k t + m/2)k λ k! k! 1 1 ) m + u k λ k! + j=0 + j=0 m + j 1 ) j) k j λ j m + j 1 ) t + j + m/2)) k. j λ j Here, we ave used te series expansions for te geometric progression and te exponential function, as well as te intensive applications of Fubini s teorem and Caucy teorem on te product of series. Ten we conclude tat μ t k, = 1 1 ) m + m + j 1 ) t + j + m/2)) k, k N λ j λ j 0, j=0 wic finises te proof in te case λ > 1. Using a completely similar argumentation we prove te result for λ <1.