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Acknowledgements The authors would like to express their gratitude to the two anonymous reviewers for their useful comments and suggestions, which improved the comprehension of the manuscript. In particular, we thank the reviewer who pointed out references [4–6, 15]., Funding for open access charge: Universidad de Granada / CBUA This research was supported through the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES), in the scope of the CAPES-PrInt Program, process number 88887.310463/2018-00, International Cooperation Project number 88887.468471/2019-00. The second author (MAP) has been partially supported by grant PGC2018-094932-B-I00 from FEDER/Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de Investigación, and the IMAG-María de Maeztu grant CEX2020-001105-M/ AEI/10.13039/501100011033., We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i−1 , where ω(t) is an univariate weight function in t∈(a,b) and x=(x1,x2,…,xd) with xi∈(a,b). Applying the change of variables xi, i=1,2,…,d, into ur, r=1,2,…,d, where ur is the r-th elementary symmetric function, we obtain the domain region in terms of the discriminant of the polynomials having xi, i=1,2,…,d, as its zeros and in terms of the corresponding Sturm sequence. Choosing the univariate weight function as the Hermite, Laguerre, and Jacobi weight functions, we obtain the representation in terms of the variables ur for the partial differential operators such that the respective Hermite, Laguerre, and Jacobi generalized multivariate orthogonal polynomials are the eigenfunctions. Finally, we present explicitly the partial differential operators for Hermite, Laguerre, and Jacobi generalized polynomials, for d=2 and d=3 variables., Funding for open access charge: Universidad de Granada / CBUA, Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES), in the scope of the CAPES-PrInt Program, process number 88887.310463/2018-00, International Cooperation Project number 88887.468471/2019-00, Grant PGC2018-094932-B-I00 from FEDER/Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de Investigación, IMAG-María de Maeztu grant CEX2020-001105-M/ AEI/10.13039/501100011033

We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special $R_{II}$ recurrence relation. We also look into some methods for generating the nodes (which lie on the real line) and the positive weights of these quadrature rules. With a simple transformation these quadrature rules on the real line also lead to certain positive quadrature rules of highest algebraic degree of precision on the unit circle. This way, we also introduce new approaches to evaluate the nodes and weights of these specific quadrature rules on the unit circle., Comment: 32 pages

We study an energy-dependent potential related to the Rosen&ndash, Morse potential. We give in closed-form the expression of a system of eigenfunctions of the Schr&ouml, dinger operator in terms of a class of functions associated to a family of hypergeometric para-orthogonal polynomials on the unit circle. We also present modified relations of orthogonality and an asymptotic formula. Consequently, bound state solutions can be obtained for some values of the parameters that define the model. As a particular case, we obtain the symmetric trigonometric Rosen&ndash, Morse potential for which there exists an orthogonal basis of eigenstates in a Hilbert space. By comparing the existent solutions for the symmetric trigonometric Rosen&ndash, Morse potential, an identity involving Gegenbauer polynomials is obtained.

When a measure $\varPsi(x)$ on the real line is subjected to the modification $d\varPsi^{(t)}(x) = e^{-tx} d \varPsi(x)$ , then the coefficients of the recurrence relation of the orthogonal polynomials in $x$ with respect to the measure $\varPsi^{(t)}(x)$ are known to satisfy the so-called Toda lattice formulas as functions of $t$ . In this paper we consider a modification of the form $e^{-t(\mathfrak{p}x+ \mathfrak{q}/x)}$ of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either $\mathfrak{p}=0$ or $\mathfrak{q}=0$ . However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.

It is known that given a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, we can associate a unique nontrivial probability measure μ on the unit circle. Precisely, the measure is such that the corresponding Verblunsky coefficients {αn}n=0∞ are given by the relation αn−1=ρ‾n−1[1−2mn−icn1−icn],n≥1, where ρ0=1ρ0=1, ρn=∏k=1n(1−ick)/(1+ick), n≥1n≥1 and {mn}n=0∞ is the minimal parameter sequence of {dn}n=1∞. In this paper we consider the space, denoted by NpNp, of all nontrivial probability measures such that the associated real sequences {cn}n=1∞ and {mn}n=1∞ are periodic with period p , for p∈Np∈N. By assuming an appropriate metric on the space of all nontrivial probability measures on the unit circle, we show that there exists a homeomorphism gpgp between the metric subspaces NpNp and VpVp, where VpVp denotes the space of nontrivial probability measures with associated p -periodic Verblunsky coefficients. Moreover, it is shown that the set FpFp of fixed points of gpgp is exactly Vp∩NpVp∩Np and this set is characterized by a (p−1)(p−1)-dimensional submanifold of RpRp. We also prove that the study of probability measures in NpNp is equivalent to the study of probability measures in VpVp. Furthermore, it is shown that the pure points of measures in NpNp are, in fact, zeros of associated para-orthogonal polynomials of degree p . We also look at the essential support of probability measures in the limit periodic case, i.e., when the sequences {cn}n=1∞ and {mn}n=1∞ are limit periodic with period p. Finally, we give some examples to illustrate the results obtained.

A result of P\'olya states that every sequence of quadrature formulas $Q_n(f)$ with $n$ nodes and positive numbers converges to the integral $I(f)$ of a continuous function $f$ provided $Q_n(f)=I(f)$ for a space of algebraic polynomials of certain degree that depends on $n$. The classical case when the algebraic degree of precision is the highest possible is well-known and the quadrature formulas are the Gaussian ones whose nodes coincide with the zeros of the corresponding orthogonal polynomials and the numbers are expressed in terms of the so-called kernel polynomials. In many cases it is reasonable to relax the requirement for the highest possible degree of precision in order to gain the possibility to either approximate integrals of more specific continuous functions that contain a polynomial factor or to include additional fixed nodes. The construction of such quadrature processes is related to quasi-orthogonal polynomials. Given a sequence $\left \{ P_{n}\right \}_{n\geq0}$ of monic orthogonal polynomials and a fixed integer $k$, we establish necessary and sufficient conditions so that the quasi-orthogonal polynomials $\left \{ Q_{n}\right \}_{n\geq0}$ defined by \[ Q_{n}(x) =P_{n}(x) + \sum \limits_{i=1}^{k-1} b_{i,n}P_{n-i}(x), \ \ n\geq 0, \] with $b_{i,n} \in \mathbb{R}$, and $b_{k-1,n}\neq 0$ for $n\geq k-1$, also constitute a sequence of orthogonal polynomials. Therefore we solve the inverse problem for linearly related orthogonal polynomials. The characterization turns out to be equivalent to some nice recurrence formulas for the coefficients $b_{i,n}$. We employ these results to establish explicit relations between various types of quadrature rules from the above relations. A number of illustrative examples are provided., Comment: 31 pages, 4 figures

It was shown recently that associated with a pair of real sequences $$\{\{c_{n}\}_{n=1}^{\infty }, \{d_{n}\}_{n=1}^{\infty }\}$$ , with $$\{d_{n}\}_{n=1}^{\infty }$$ a positive chain sequence, there exists a unique nontrivial probability measure $$\mu $$ on the unit circle. The Verblunsky coefficients $$\{\alpha _{n}\}_{n=0}^{\infty }$$ associated with the orthogonal polynomials with respect to $$\mu $$ are given by the relation $$\begin{aligned} \alpha _{n-1}=\overline{\tau }_{n-1}\left[ \frac{1-2m_{n}-ic_{n}}{1-ic_{n}}\right] , \quad n \ge 1, \end{aligned}$$ where $$\tau _0 = 1$$ , $$\tau _{n}=\prod _{k=1}^{n}(1-ic_{k})/(1+ic_{k})$$ , $$n \ge 1$$ and $$\{m_{n}\}_{n=0}^{\infty }$$ is the minimal parameter sequence of $$\{d_{n}\}_{n=1}^{\infty }$$ . In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences $$\{c_{n}\}_{n=1}^{\infty }$$ and $$\{m_{n}\}_{n=1}^{\infty }$$ . When the sequence $$ \{c_{n}\}_{n=1}^{\infty }$$ is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the measure in the neighbourhood of $$z= -1$$ . Furthermore, we show that it is possible to generate periodic Verblunsky coefficients by choosing periodic sequences $$\{c_{n}\}_{n=1}^{\infty }$$ and $$\{m_{n}\}_{n=1}^{\infty }$$ with the additional restriction $$c_{2n}=-c_{2n-1}, \, n\ge 1.$$ We also give some results on periodic Verblunsky coefficients from the point of view of positive chain sequences. An example is provided to illustrate the results obtained.

The purpose of this paper is twofold. Firstly we investigate the distribution, simplicity and monotonicity of the zeros around the unit circle and real line of the real self-reciprocal polynomials R n ( λ ) ( z ) = 1 + λ ( z + z 2 + ⋯ + z n − 1 ) + z n , n ≥ 2 and λ ∈ R . Secondly, as an application of the first results we give necessary and sufficient conditions to guarantee that all zeros of the self-reciprocal polynomials S n ( λ ) ( z ) = ∑ k = 0 n s n , k ( λ ) z k , n ≥ 2 , with s n , 0 ( λ ) = s n , n ( λ ) = 1 , s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , ⌊ n / 2 ⌋ when n is odd, and s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , n / 2 − 1 , s n , n / 2 ( λ ) = ( n / 2 ) λ when n is even, lie on the unit circle, solving then an open problem given by Kim and Park in 2008.

Nos ultimos anos os polinomios ortogonais no circulo unitario, tambem conhecidos como polinomios de Szego, vem recebendo bastante atencao de muitos pesquisadores. Este fato tem ocorrido principalmente em razao de suas aplicacoes em diversas areas da Matematica. Regras de quadratura, processamento de sinais e teoria espectral sao alguns dos muitos topicos em que estes polinomios estao inseridos (veja, por exemplo, [1, 6]).

Given a nontrivial positive measure μ on the unit circle T , the associated Christoffel–Darboux kernels are K n ( z , w ; μ ) = ∑ k = 0 n φ k ( w ; μ ) ¯ φ k ( z ; μ ) , n ≥ 0 , where φ k ( ⋅ ; μ ) are the orthonormal polynomials with respect to the measure μ . Let the positive measure ν on the unit circle be given by d ν ( z ) = | G 2 m ( z ) | d μ ( z ) , where G 2 m is a conjugate reciprocal polynomial of exact degree 2 m . We establish a determinantal formula expressing { K n ( z , w ; ν ) } n ≥ 0 directly in terms of { K n ( z , w ; μ ) } n ≥ 0 . Furthermore, we consider the special case of w = 1 ; it is known that appropriately normalized polynomials K n ( z , 1 ; μ ) satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters { c n ( μ ) } n = 1 ∞ and { g n ( μ ) } n = 1 ∞ , with 0 g n 1 for n ≥ 1 . The double sequence { ( c n ( μ ) , g n ( μ ) ) } n = 1 ∞ characterizes the measure μ . A natural question about the relation between the parameters c n ( μ ) , g n ( μ ) , associated with μ , and the sequences c n ( ν ) , g n ( ν ) , corresponding to ν , is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of T ), a measure for which the Christoffel–Darboux kernels, with w = 1 , are given by basic hypergeometric polynomials and a measure for which the orthogonal polynomials and the Christoffel–Darboux kernels, again with w = 1 , are given by hypergeometric polynomials.

Classical orthogonal polynomials can be characterized in terms of the corresponding Stieltjes function.We consider the construction of a Stieltjes function in terms of the falling factorials for discrete classical orthogonal polynomials (Charlier, Krawtchouk, Meixner, and Hahn). This Stieltjes function associated with classical orthogonal polynomials of a discrete variable is solution of a non-homogeneous difference equation. That property characterizes the discrete classical measures. In addition, an hypergeometric expression for the Stieltjes function is obtained in all the discrete classical cases.

Sobolev orthogonal polynomials in two variables are defined via inner products involving gradients. Such a kind of inner product appears in connection with several physical and technical problems. Matrix second-order partial differential equations satisfied by Sobolev orthogonal polynomials are studied. In particular, we explore the connection between the coefficients of the second-order partial differential operator and the moment functionals defining the Sobolev inner product. Finally, some old and new examples are given.

We consider the sequence of polynomials {Q n } satisfying the L-orthogonality ?[z ?n+m Q n (z)]=0, 0?m?n?1, with respect to a linear functional ? for which the moments ?[t n ]=μ n are all complex. Under certain restriction on the moment functional these polynomials also satisfy a three term recurrence relation. We consider three special classes of such moment functionals and characterize them in terms of the coefficients of the associated three term recurrence relations. Relations between the polynomials {Q n } associated with two of these special classes of moment functionals are also given. Examples are provided to justify this characterization.

Iserles et al. (J. Approx. Theory 65:151–175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt (J. Approx. Theory 114:115–140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt, when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair.

Inner products of the type f , gS = f , gψ0 + f , gψ1 , where one of the measures ψ0 or ψ1 is the measure associated with the Gegenbauer polynomials, are usually referred to as Gegenbauer-Sobolev inner products. This paper deals with some asymptotic relations for the orthogonal polynomials with respect to a class of Gegenbauer-Sobolev inner products. The inner products are such that the associated pairs of symmetric measures (ψ0, ψ1) are not within the concept of symmetrically coherent pairs of measures.

Resumo. Recentemente, foi mostrado em [5] que associado ao par de sequencias reais {{cn}n=1, {dn}n=1}, com {dn}n=1 uma sequencia encadeada positiva, existe uma unica medida de probabilidade nao trivial μ no circulo unitario. Mostrou-se tambem que os coeficientes de Verblunsky {αn}n=0, associados aos polinomios ortogonais com respeito a μ, podem ser relacionados diretamente com tais sequencias. Neste trabalho, consideramos esta relacao e suas consequencias quando impomos uma restricao de sinal sobre a sequencia {cn}n=1. Precisamente, quando a sequencia {cn}n=1 tem uma propriedade de sinal alternante, usamos informacoes sobre os zeros de certos polinomios para-ortogonais para estimar o suporte da medida associada.

onde b ∈ R \ Z−. Observe que se p = 1 (resp. q = 1), entao αn (resp. βn) nao aparece na expressao de pFq, por exemplo para p = 1 e q = 1 temos o polinomio 1F1(−n; b; z). ∗Pesquisa desenvolvida no programa “Research in Pairs” do Mathematisches Forschungsinstitut Oberwolfach. †apoio do CNPq, FUNDUNESP e FAPESP, Brasil. ‡apoio do Ministerio de Ciencia e Innovacion, Espanha e ERDF, projeto MTM2011-28952-C02-01, e da Junta de Andalucia, Espanha, Grupo de Pesquisa FQM–0229 (Campus de Excelencia Internacional CEI–MAR) projeto P09–FQM–4643. Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, Vol. 3, N. 1, 2015. Trabalho apresentado no XXXV CNMAC, Natal-RN, 2014.

Three term recurrence formula has been an important tool in the studies of orthogonal polynomials on the real line. Positive chain sequences also play an important role in these studies. In recent years we have realized that the use of a different type of three term recurrence formula in combination with the use of positive chain sequences is also very important for studying the properties of orthogonal polynomials on the unit circle. In this text we give some new connection that exists between orthogonal polynomials on the unit circle and para-orthogonal polynomials given by three term recurrence formula.

Neste trabalho, apresenta-se resultados sobre a localizacao dos zeros dos polinomios de Szego com coeficientes de reflexao maiores que um em modulo. Exemplos numericos sao apresentados para ilustrar os resultados .

Os polinomios ortogonais em uma variavel sao ferramentas importantes na solucao de diversos tipos de problemas e sua teoria contribui nos estudos relacionados a estabilidade numerica, equacoes diferenciais, fracoes continuas, teoria da aproximacao, entre outros. A teoria desses polinomios e amplamente estudada, com muitos trabalhos publicados na area, como o livro de T. S. Chihara [2]. Em varias variaveis, os estudos desses polinomios tem-se difundido com maior intensidade nas ultimas decadas. Segundo C. F. Dunkl e Y. Xu em [3], o primeiro trabalho nessa area e o livro [1] “Fonctions Hypergeometriques et Hyperspheriques Polynomes D’Hermite” de P. Appell e J. Kampe de Feriet, de 1926, que foi tomado como base neste trabalho. De acordo com [3], os poucos livros dedicados a teoria geral dos polinomios ortogonais em varias variaveis tem como enfase o tipo classico desses polinomios, ou seja, as familias de polinomios cujas funcoes peso tem como dominio as regioes regulares: o quadrado, o simplex, a bola em Rn ou o proprio Rn. Aqui, apresentamos como os polinomios de Legendre em n variaveis, de grau m1 + · · · + mn, denotados por Vm1,...,mn(x1, ..., xn), que sao ortogonais na regiao da bola unitaria, e algumas de suas propriedades, podem ser obtidas atraves da extensao de alguns de seus conceitos e propriedades destes polinomios ja conhecidas em uma variavel. Um exemplo desta extensao e a funcao hipergeometrica, que em uma variavel e dada, para Vm(x), por Vm(x) = 2 m (1 2 )

Connection between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), are looked at. The results are applied to obtain information regarding Sobolev orthogonal polynomials associated with certain pairs of measures.

Submitted by Guilherme Lemeszenski (guilherme@nead.unesp.br) on 2014-02-26T17:15:47Z No. of bitstreams: 1 WOS000181888700008.pdf: 150681 bytes, checksum: 728da2bf670384f99c93e88449528e28 (MD5) Made available in DSpace on 2014-02-26T17:15:47Z (GMT). No. of bitstreams: 1 WOS000181888700008.pdf: 150681 bytes, checksum: 728da2bf670384f99c93e88449528e28 (MD5) Previous issue date: 2003-04-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T15:20:39Z No. of bitstreams: 1 WOS000181888700008.pdf: 150681 bytes, checksum: 728da2bf670384f99c93e88449528e28 (MD5) Made available in DSpace on 2014-05-20T15:20:39Z (GMT). No. of bitstreams: 1 WOS000181888700008.pdf: 150681 bytes, checksum: 728da2bf670384f99c93e88449528e28 (MD5) Previous issue date: 2003-04-01 We consider the real Szego polynomials and obtain some relations to certain self inversive orthogonal L-polynomials defined on the unit circle and corresponding symmetric orthogonal polynomials on real intervals. We also consider the polynomials obtained when the coefficients in the recurrence relations satisfied by the self inversive orthogonal L-polynomials are rotated. (C) 2002 Elsevier B.V. B.V. All rights reserved. Univ Estadual Paulista, Dept Ciências Comp & Estatist, Inst Biociencias Letras & Ciências Exatas, BR-15054000 Sao Jose do Rio Preto, SP, Brazil Univ Estadual Paulista, Dept Ciências Comp & Estatist, Inst Biociencias Letras & Ciências Exatas, BR-15054000 Sao Jose do Rio Preto, SP, Brazil

Submitted by Guilherme Lemeszenski (guilherme@nead.unesp.br) on 2014-02-26T17:13:36Z No. of bitstreams: 1 WOS000176146300007.pdf: 132901 bytes, checksum: fc1b950412460776d2f1efc9b5384394 (MD5) Made available in DSpace on 2014-02-26T17:13:36Z (GMT). No. of bitstreams: 1 WOS000176146300007.pdf: 132901 bytes, checksum: fc1b950412460776d2f1efc9b5384394 (MD5) Previous issue date: 2002-06-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T15:26:54Z No. of bitstreams: 1 WOS000176146300007.pdf: 132901 bytes, checksum: fc1b950412460776d2f1efc9b5384394 (MD5) Made available in DSpace on 2014-05-20T15:26:54Z (GMT). No. of bitstreams: 1 WOS000176146300007.pdf: 132901 bytes, checksum: fc1b950412460776d2f1efc9b5384394 (MD5) Previous issue date: 2002-06-01 in this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K-n((lambda.,M,k)) associated with the probability measure dphi(lambda,M,k;x), which is the Gegenbauer measure of parameter lambda + 1 with two additional mass points at +/-k. When k = 1 we obtain information on the polynomials K-n((lambda.,M)) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K-n((lambda,M,k)) in relation to M and k are also given. (C) 2002 Elsevier B.V. B.V. All rights reserved. Univ Estadual Paulista, Inst Biociencias Letras & Ciências Exatas, Dept Ciências Comp & Estat, BR-15054000 Sao Jose do Rio Preto, SP, Brazil Univ Estadual Paulista, Inst Biociencias Letras & Ciências Exatas, Dept Ciências Comp & Estat, BR-15054000 Sao Jose do Rio Preto, SP, Brazil

Made available in DSpace on 2018-12-11T17:03:38Z (GMT). No. of bitstreams: 0 Previous issue date: 2016-11-01 When a nontrivial measure μ on the unit circle satisfies the symmetry dμ(ei(2π-θ))=-dμ(eiθ) then the associated orthogonal polynomials on the unit circle, say Φn, are all real. In this case, in 1986, Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zΦn(z)+Φn∗(z)} and {zΦn(z)-Φn∗(z)}, where Φn∗(z)=zn Φn(1/z/)/, satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [-1,1]. The same authors, in 1988, have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently the extension associated with the para-orthogonal polynomials zΦn(z)-Φn∗(z) was thoroughly explored, especially from the point of view of three term recurrence and chain sequences. The main objective of the present article is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zΦn(z)+Φn∗(z) for any nontrivial measure on the unit circle. As an important application of the theory, a characterization for the existence of the integral ∫02π|eiθ-w|-2dμ(eiθ), where w is such that |w|=1, is given in terms of the coefficients αn-1=-Φn(0)/, n≥1. Examples are also provided to justify all the results. Departamento de Matemática Aplicada IBILCE UNESP - Univ. Estadual Paulista Department of Mathematics IIT Roorkee Departamento de Matemática Aplicada IBILCE UNESP - Univ. Estadual Paulista

In this paper, we consider the symmetric Gaussian and L-Gaussian quadrature rules associated with twin periodic recurrence relations with possible variations in the initial coefficient. We show that the weights of the associated Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 4. We also show that the weights of the associated L-Gaussian quadrature rules can be given as rational functions in terms of the corresponding nodes where the numerators and denominators are polynomials of degree at most 5. Special cases of these quadrature rules are given. Finally, an easy to implement procedure for the evaluation of the nodes is described.

We consider interpolatory quadrature rules with nodes and weights satisfying symmetric properties in terms of the division operator. Information concerning these quadrature rules is obtained using a transformation that exists between these rules and classical symmetric interpolatory quadrature rules. In particular, we study those interpolatory quadrature rules with two fixed nodes. We obtain specific examples of such quadrature rules.

Submitted by Guilherme Lemeszenski (guilherme@nead.unesp.br) on 2014-02-26T17:03:01Z No. of bitstreams: 1 WOS000080681500014.pdf: 112775 bytes, checksum: e7120be31e58aa5eecc4f6787b38d593 (MD5) Made available in DSpace on 2014-02-26T17:03:01Z (GMT). No. of bitstreams: 1 WOS000080681500014.pdf: 112775 bytes, checksum: e7120be31e58aa5eecc4f6787b38d593 (MD5) Previous issue date: 1999-05-31 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T14:01:35Z No. of bitstreams: 1 WOS000080681500014.pdf: 112775 bytes, checksum: e7120be31e58aa5eecc4f6787b38d593 (MD5) Made available in DSpace on 2014-05-20T14:01:35Z (GMT). No. of bitstreams: 1 WOS000080681500014.pdf: 112775 bytes, checksum: e7120be31e58aa5eecc4f6787b38d593 (MD5) Previous issue date: 1999-05-31 A strong Stieltjes distribution d psi(t) is called symmetric if it satisfies the propertyt(omega) d psi(beta(2)/t) = -(beta(2)/t)(omega) d psi(t), for t is an element of (a, b) subset of or equal to (0, infinity), 2 omega is an element of Z, and beta > 0.In this article some consequences of symmetry on the moments, the orthogonal L-polynomials and the quadrature formulae associated with the distribution are given. (C) 1999 Elsevier B.V. B.V. All rights reserved. UNESP, IBILCE, Dept Ciências Comp & Estatist, BR-15054000 Sao Jose Rio Preto, SP, Brazil Univ St Andrews, Sch Math & Computat Sci, St Andrews KY16 9SS, Fife, Scotland UNESP, IBILCE, Dept Ciências Comp & Estatist, BR-15054000 Sao Jose Rio Preto, SP, Brazil

The main goal in this manuscript is to present a class of functions satisfying a certain orthogonality property for which there also exists a three term recurrence formula. This class of functions, which can be considered as an extension to the class of symmetric orthogonal polynomials on $[-1,1]$, has a complete connection to the orthogonal polynomials on the unit circle. Quadrature rules and other properties based on the zeros of these functions are also considered., Comment: 20 pages, 2 figures

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:25Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:47:08Z : No. of bitstreams: 1 2-s2.0-78649931864.pdf: 245342 bytes, checksum: be68218069daea10a04db8be16aabf97 (MD5) Made available in DSpace on 2014-05-27T11:25:25Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-12-14 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Ministerio de Ciência e Innovacion (Micinn) of Spain European Regional Development Fund (ERDF) Junta de Andalucia Zeros of orthogonal polynomials associated with two different Sobolev inner products involving the Jacobi measure are studied. In particular, each of these Sobolev inner products involves a pair of closely related Jacobi measures. The measures of the inner products considered are beyond the concept of coherent pairs of measures. Existence, real character, location and interlacing properties for the zeros of these Jacobi-Sobolev orthogonal polynomials are deduced. © 2010 SBMAC. DCCE IBILCE UNESP - Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de Matemática Aplicada Instituto Carlos I de Física Teórica y Computacional Universidad de Granada, 18071 Granada DCCE IBILCE UNESP - Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP European Regional Development Fund: MTM-2008-06689-C02-02 Junta de Andalucia: G.I. FQM 0229

Submitted by Guilherme Lemeszenski (guilherme@nead.unesp.br) on 2014-02-26T17:29:44Z No. of bitstreams: 1 WOS000247015800009.pdf: 215971 bytes, checksum: 8a917ecf6dc070f2ec5053c13a84e3ff (MD5) Made available in DSpace on 2014-02-26T17:29:44Z (GMT). No. of bitstreams: 1 WOS000247015800009.pdf: 215971 bytes, checksum: 8a917ecf6dc070f2ec5053c13a84e3ff (MD5) Previous issue date: 2007-06-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T15:23:20Z No. of bitstreams: 1 WOS000247015800009.pdf: 215971 bytes, checksum: 8a917ecf6dc070f2ec5053c13a84e3ff (MD5) Made available in DSpace on 2014-05-20T15:23:20Z (GMT). No. of bitstreams: 1 WOS000247015800009.pdf: 215971 bytes, checksum: 8a917ecf6dc070f2ec5053c13a84e3ff (MD5) Previous issue date: 2007-06-01 We consider a connection that exists between orthogonal polynomials associated with positive measures on the real line and orthogonal Laurent polynomials associated with strong measures of the class S-3 [0, beta, b]. Examples are given to illustrate the main contribution in this paper. (c) 2006 Elsevier B.V. All rights reserved. Univ Estadual Paulista, UNESP, DCCE, IBILCE, BR-15054000 Sao Jose do Rio Preto, SP, Brazil Univ Estadual Paulista, UNESP, DCCE, IBILCE, BR-15054000 Sao Jose do Rio Preto, SP, Brazil

Submitted by Guilherme Lemeszenski (guilherme@nead.unesp.br) on 2014-02-26T17:20:17Z No. of bitstreams: 0 Made available in DSpace on 2014-02-26T17:20:17Z (GMT). No. of bitstreams: 0 Previous issue date: 2004-01-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T14:01:31Z No. of bitstreams: 0 Made available in DSpace on 2014-05-20T14:01:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2004-01-01 We study the use of para-orthogonal polynomials in solving the frequency analysis problem. Through a transformation of Delsarte and Genin, we present an approach for the frequency analysis by using the zeros and Christoffel numbers of polynomials orthogonal on the real line. This leads to a simple and fast algorithm for the estimation of frequencies. We also provide a new method, faster than the Levinson algorithm, for the determination of the reflection coefficients of the corresponding real Szego polynomials from the given moments. Univ Estadual Paulista, IBILCE, Dept Ciências Computacao & Estatist, BR-15054000 São Paulo, Brazil Univ Cent Florida, Dept Math, Orlando, FL 32816 USA Univ Estadual Paulista, IBILCE, Dept Ciências Computacao & Estatist, BR-15054000 São Paulo, Brazil

We consider the Sobolev inner product [email protected]!"-"1^1f(x)g(x)(1-x^2)^@a^-^1^[email protected]!f^'(x)g^'(x)[email protected](x),@a>-12, where [email protected] is a measure involving a Gegenbauer weight and with mass points outside the interval (-1,1). We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product. We obtain the asymptotics of the largest zeros of these polynomials via a Mehler-Heine type formula. These results are illustrated with some numerical experiments.

Submitted by Guilherme Lemeszenski (guilherme@nead.unesp.br) on 2014-02-26T17:22:01Z No. of bitstreams: 1 WOS000229137200004.pdf: 244766 bytes, checksum: 4f948f1945375dd42e7f57ba2d95e455 (MD5) Made available in DSpace on 2014-02-26T17:22:01Z (GMT). No. of bitstreams: 1 WOS000229137200004.pdf: 244766 bytes, checksum: 4f948f1945375dd42e7f57ba2d95e455 (MD5) Previous issue date: 2005-07-01 Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-20T15:22:52Z No. of bitstreams: 1 WOS000229137200004.pdf: 244766 bytes, checksum: 4f948f1945375dd42e7f57ba2d95e455 (MD5) Made available in DSpace on 2014-05-20T15:22:52Z (GMT). No. of bitstreams: 1 WOS000229137200004.pdf: 244766 bytes, checksum: 4f948f1945375dd42e7f57ba2d95e455 (MD5) Previous issue date: 2005-07-01 This paper deals with the classes S-3(omega, beta, b) of strong distribution functions defined on the interval [beta(2)/b, b], 0 < beta < b

Made available in DSpace on 2018-12-11T16:46:02Z (GMT). No. of bitstreams: 0 Previous issue date: 2017-11-01 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) A method is presented for the calculation of roots of non-polynomial functions, motivated by the requirement to generate quadrature rules based on non-polynomial orthogonal functions. The approach uses a combination of local Taylor expansions and Sturm’s theorem for roots of a polynomial which together give a means of efficiently generating estimates of zeros which can be polished using Newton’s method. The technique is tested on a number of realistic problems including some chosen to be highly oscillatory and to have large variations in amplitude, both of which features pose particular challenges to root–finding methods. Departamento de Matemática Aplicada UNESP–University Estadual Paulista Department of Mechanical Engineering University of Bath Departamento de Matemática Aplicada UNESP–University Estadual Paulista FAPESP: 2014/17357-1 FAPESP: 2014/22571-2