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Journal articles on the topic 'Orthogonal polynomials'

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Published: 4 June 2021

Last updated: 29 July 2024

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1

Lytvynov, Eugene, and Irina Rodionova. "Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise." Infinite Dimensional Analysis, Quantum Probability and Related Topics 21, no. 02 (June 2018): 1850011. http://dx.doi.org/10.1142/s021902571850011x.

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Let [Formula: see text] denote a non-commutative monotone Lévy process. Let [Formula: see text] denote the corresponding monotone Lévy noise, i.e. formally [Formula: see text]. A continuous polynomial of [Formula: see text] is an element of the corresponding non-commutative [Formula: see text]-space [Formula: see text] that has the form [Formula: see text], where [Formula: see text]. We denote by [Formula: see text] the space of all continuous polynomials of [Formula: see text]. For [Formula: see text], the orthogonal polynomial [Formula: see text] is defined as the orthogonal projection of the monomial [Formula: see text] onto the subspace of [Formula: see text] that is orthogonal to all continuous polynomials of [Formula: see text] of order [Formula: see text]. We denote by [Formula: see text] the linear span of the orthogonal polynomials. Each orthogonal polynomial [Formula: see text] depends only on the restriction of the function [Formula: see text] to the set [Formula: see text]. The orthogonal polynomials allow us to construct a unitary operator [Formula: see text], where [Formula: see text] is an extended monotone Fock space. Thus, we may think of the monotone noise [Formula: see text] as a distribution of linear operators acting in [Formula: see text]. We say that the orthogonal polynomials belong to the Meixner class if [Formula: see text]. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: [Formula: see text] and [Formula: see text]. In this case, the monotone Lévy noise has the representation [Formula: see text]. Here, [Formula: see text] and [Formula: see text] are the (formal) creation and annihilation operators at [Formula: see text] acting in [Formula: see text].

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2

Bihun, Oksana, and Clark Mourning. "Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials." Advances in Mathematical Physics 2018 (2018): 1–10. http://dx.doi.org/10.1155/2018/4710754.

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Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family pνxν=0∞ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations Apν(x)=qν(x)pν(x), where A is a linear differential operator and each qν(x) is a polynomial of degree at most n0∈N; n0 does not depend on ν. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.

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3

Duran, Antonio J. "Markov's Theorem for Orthogonal Matrix Polynomials." Canadian Journal of Mathematics 48, no. 6 (December 1, 1996): 1180–95. http://dx.doi.org/10.4153/cjm-1996-062-4.

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AbstractMarkov's Theorem shows asymptotic behavior of the ratio between the n-th orthonormal polynomial with respect to a positive measure and the n-th polynomial of the second kind. In this paper we extend Markov's Theorem for orthogonal matrix polynomials.

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4

Sultanakhmedov, M. S. "RECURRENCE RELATIONS FOR SOBOLEV ORTHOGONAL POLYNOMIALS." Issues of Analysis 27, no. 2 (June 2020): 97–118. http://dx.doi.org/10.15393/j3.art.2020.7290.

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5

Farahmand, K. "On random orthogonal polynomials." Journal of Applied Mathematics and Stochastic Analysis 14, no. 3 (January 1, 2001): 265–74. http://dx.doi.org/10.1155/s1048953301000223.

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Let T0∗(x),T1∗(x),…,Tn∗(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (−1,1). The asymptotic estimate of the expected number of real zeros of the random polynomial g0T0∗(x)+g1T1∗(x)+…+gnTn∗(x) where gj, j=1,2,…,n are independent identically and normally distributed random variables is known. In this paper, we first present the asymptotic value for the above expected number when coefficients are dependent random variables. Further, for the case of independent coefficients, we define the expected number of zero up-crossings with slope greater than u or zero down-crossings with slope less than −u. Promoted by the graphical interpretation, we define these crossings as u-sharp. For the above polynomial, we provide the expected number of such crossings.

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6

Ranga, A. Sri. "Symmetric Orthogonal Polynomials and the Associated Orthogonal L-Polynomials." Proceedings of the American Mathematical Society 123, no. 10 (October 1995): 3135. http://dx.doi.org/10.2307/2160672.

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7

Lee, Seung-Yeop, and Meng Yang. "Planar orthogonal polynomials as Type II multiple orthogonal polynomials." Journal of Physics A: Mathematical and Theoretical 52, no. 27 (June 7, 2019): 275202. http://dx.doi.org/10.1088/1751-8121/ab1af9.

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8

Sri Ranga, A. "Symmetric orthogonal polynomials and the associated orthogonal $L$-polynomials." Proceedings of the American Mathematical Society 123, no. 10 (October 1, 1995): 3135. http://dx.doi.org/10.1090/s0002-9939-1995-1291791-7.

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9

de Andrade, E. X. L., C. F. Bracciali, and A. Sri Ranga. "Another connection between orthogonal polynomials and L-orthogonal polynomials." Journal of Mathematical Analysis and Applications 330, no. 1 (June 2007): 114–32. http://dx.doi.org/10.1016/j.jmaa.2006.07.012.

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10

Charris, Jairo A., and Mourad E. H. Ismail. "Sieved Orthogonal Polynomials. VII: Generalized Polynomial Mappings." Transactions of the American Mathematical Society 340, no. 1 (November 1993): 71. http://dx.doi.org/10.2307/2154546.

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11

Szabłowski, Paweł J. "Markov processes, polynomial martingales and orthogonal polynomials." Stochastics 90, no. 1 (April 10, 2017): 61–77. http://dx.doi.org/10.1080/17442508.2017.1311899.

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12

Charris, Jairo A., and Mourad E. H. Ismail. "Sieved orthogonal polynomials. VII. Generalized polynomial mappings." Transactions of the American Mathematical Society 340, no. 1 (January 1, 1993): 71–93. http://dx.doi.org/10.1090/s0002-9947-1993-1038014-4.

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13

de Jesus, M. N., and J. Petronilho. "On orthogonal polynomials obtained via polynomial mappings." Journal of Approximation Theory 162, no. 12 (December 2010): 2243–77. http://dx.doi.org/10.1016/j.jat.2010.07.012.

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14

Castillo, K., M. N. de Jesus, and J. Petronilho. "On semiclassical orthogonal polynomials via polynomial mappings." Journal of Mathematical Analysis and Applications 455, no. 2 (November 2017): 1801–21. http://dx.doi.org/10.1016/j.jmaa.2017.06.072.

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15

Stojanovic, Nikola, and Negovan Stamenkovic. "Lowpass filters approximation based on the Jacobi polynomials." Facta universitatis - series: Electronics and Energetics 30, no. 3 (2017): 351–62. http://dx.doi.org/10.2298/fuee1703351s.

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A case study related to the design the analog lowpass filter using a set of orthogonal Jacobi polynomials, having four parameters to vary, is considered. The Jacobi polynomial has been modified in order to be used as a filter approximating function. The obtained magnitude response is more general than the response of the classical ultra-spherical filter, due to one additional parameter available in orthogonal Jacobi polynomials. This additional parameter may be used to obtain a magnitude response having either smaller passband ripple, smaller group delay variation or sharper cutoff slope. Two methods for transfer function approximation are investigated: the first method is based on the known shifted Jacobi polynomial, and the second method is based on the proposed modification of Jacobi polynomials. The shifted Jacobi polynomials are suitable only for odd degree transfer function. However, the proposed modified Jacobi polynomial filter function is more general but not orthogonal. It is transformed into orthogonal polynomial when orders are equal and then includes the Chebyshev filter of the first kind, the Chebyshev filter of the second kind, the Legendre filter, Gegenbauer (ultraspherical) filter and many other filters, as its special cases.

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16

Loureiro, Ana F., and Walter Van Assche. "Three-fold symmetric Hahn-classical multiple orthogonal polynomials." Analysis and Applications 18, no. 02 (July 17, 2019): 271–332. http://dx.doi.org/10.1142/s0219530519500106.

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We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as [Formula: see text]-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a [Formula: see text]-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them [Formula: see text]-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

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17

Liu, Rong. "Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)." Open Mathematics 18, no. 1 (March 10, 2020): 138–49. http://dx.doi.org/10.1515/math-2020-0011.

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Abstract Let Wα,ρ = xα(1 – x2)ρe–Q(x), where α > – $\begin{array}{} \displaystyle \frac12 \end{array}$ and Q is continuous and increasing on [0, 1), with limit ∞ at 1. This paper deals with orthogonal polynomials for the weights $\begin{array}{} \displaystyle W^2_{\alpha, \rho} \end{array}$ and gives bounds on orthogonal polynomials, zeros, Christoffel functions and Markov inequalities. In addition, estimates of fundamental polynomials of Lagrange interpolation at the zeros of the orthogonal polynomial and restricted range inequalities are obtained.

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18

Czyżycki, Tomasz, Jiří Hrivnák, and Jiří Patera. "Generating Functions for Orthogonal Polynomials of A2, C2 and G2." Symmetry 10, no. 8 (August 20, 2018): 354. http://dx.doi.org/10.3390/sym10080354.

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The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

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19

Rajkovic, Predrag, Sladjana Marinkovic, and Miomir Stankovic. "Orthogonal polynomials with varying weight of Laguerre type." Filomat 29, no. 5 (2015): 1053–62. http://dx.doi.org/10.2298/fil1505053r.

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In this paper, we define and examine a new functional product in the space of real polynomials. This product includes the weight function which depends on degrees of the participants. In spite of it does not have all properties of an inner product, we construct the sequence of orthogonal polynomials. These polynomials can be eigenfunctions of a differential equation what was used in some considerations in the theoretical physics. In special, we consider Laguerre type weight function and prove that the corresponding orthogonal polynomial sequence is connected with Laguerre polynomials. We study their differential properties and orthogonal properties of some related rational and exponential functions.

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20

Gautschi, Walter, and Shikang Li. "A set of orthogonal polynomials induced by a given orthogonal polynomial." Aequationes Mathematicae 46, no. 1-2 (August 1993): 174–98. http://dx.doi.org/10.1007/bf01834006.

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21

Bracciali, Cleonice, Francisco Marcellán, and Serhan Varma. "Orthogonality of quasi-orthogonal polynomials." Filomat 32, no. 20 (2018): 6953–77. http://dx.doi.org/10.2298/fil1820953b.

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A result of P?lya states that every sequence of quadrature formulas Qn(f) with n nodes and positive Cotes numbers converges to the integral I(f) of a continuous function f provided Qn(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 Cotes (Christoffel) 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 {Pn}n?0 of monic orthogonal polynomials and a fixed integer k, we establish necessary and sufficient conditions so that the quasi-orthogonal polynomials {Qn}n?0 defined by Qn(x) = Pn(x) + ?k-1,i=1 bi,nPn-i(x), n ? 0, with bi,n ? R, and bk-1,n ? 0 for n ? 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 bi,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.

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22

Ftorek, Branislav, and Pavol Oršansky. "Korous Type Inequalities for Orthogonal Polynomials in two Variables." Tatra Mountains Mathematical Publications 58, no. 1 (March 1, 2014): 1–12. http://dx.doi.org/10.2478/tmmp-2014-0001.

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ABSTRACT J. Korous reached an important result for general orthogonal polynomials in one variable. He dealt with the boundedness and uniform boundedness of polynomials { Pn(x)}∞n=0 orthonormal with the weight function h(x) = δ(x) ̃h(x), where ̃h(x) is the weight function of another system of polynomials { ̃Pn(x) }∞n=0 orthonormal in the same interval and δ(x) ≥ δ0 > 0 is a certain function. We generalize this result for orthogonal polynomials in two variables multiplying their weight function h(x, y) by a polynomial, dividing h(x, y) by a polynomial, and multiplying h(x, y) with separated variables by a certain function δ(x, y).

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23

ZHOU, JIAN-RONG, SHUAI-XIA XU, and YU-QIU ZHAO. "UNIFORM ASYMPTOTICS OF A SYSTEM OF SZEGÖ CLASS POLYNOMIALS VIA THE RIEMANN–HILBERT APPROACH." Analysis and Applications 09, no. 04 (October 2011): 447–80. http://dx.doi.org/10.1142/s0219530511001947.

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We study the uniform asymptotics of a system of polynomials orthogonal on [-1, 1] with weight function w(x) = exp {-1/(1 - x2)μ}, 0 < μ < 1/2, via the Riemann–Hilbert approach. These polynomials belong to the Szegö class. In some earlier literature involving Szegö class weights, Bessel-type parametrices at the endpoints ±1 are used to study the uniform large degree asymptotics. Yet in the present investigation, we show that the original endpoints ±1 of the orthogonal interval are to be shifted to the MRS numbers ±βn, depending on the polynomial degree n and serving as turning points. The parametrices at ±βn are constructed in shrinking neighborhoods of size 1 - βn, in terms of the Airy function. The polynomials exhibit a singular behavior as compared with the classical orthogonal polynomials, in aspects such as the location of the extreme zeros, and the approximation away from the orthogonal interval. The singular behavior resembles that of the typical non-Szegö class polynomials, cf. the Pollaczek polynomials. Asymptotic approximations are obtained in overlapping regions which cover the whole complex plane. Several large-n asymptotic formulas for πn(1), i.e. the value of the nth monic polynomial at 1, and for the leading and recurrence coefficients, are also derived.

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24

Adler, M., and P. van Moerbeke. "and orthogonal polynomials." Duke Mathematical Journal 80, no. 3 (December 1995): 863–911. http://dx.doi.org/10.1215/s0012-7094-95-08029-6.

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25

Felder, Giovanni, and Thomas Willwacher. "Jointly orthogonal polynomials." Journal of the London Mathematical Society 91, no. 3 (March 11, 2015): 750–68. http://dx.doi.org/10.1112/jlms/jdv006.

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26

Sri Ranga, A. "Companion orthogonal polynomials." Journal of Computational and Applied Mathematics 75, no. 1 (November 1996): 23–33. http://dx.doi.org/10.1016/s0377-0427(96)00074-x.

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27

Aptekarev, A. I. "Multiple orthogonal polynomials." Journal of Computational and Applied Mathematics 99, no. 1-2 (November 1998): 423–47. http://dx.doi.org/10.1016/s0377-0427(98)00175-7.

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28

Nørsett, Syvert P., and Bente Østigård. "Dual-orthogonal polynomials." Numerical Algorithms 11, no. 1 (December 1996): 311–26. http://dx.doi.org/10.1007/bf02142504.

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29

Hendriksen, E., and H. van Rossum. "Orthogonal Laurent polynomials." Indagationes Mathematicae (Proceedings) 89, no. 1 (1986): 17–36. http://dx.doi.org/10.1016/1385-7258(86)90003-x.

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30

Landram, Frank G., and Bahram Alidaee. "Computing orthogonal polynomials." Computers & Operations Research 24, no. 5 (May 1997): 473–76. http://dx.doi.org/10.1016/s0305-0548(96)00071-8.

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31

Giraud, B. G. "Constrained orthogonal polynomials." Journal of Physics A: Mathematical and General 38, no. 33 (August 3, 2005): 7299–311. http://dx.doi.org/10.1088/0305-4470/38/33/006.

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32

Kroó, András. "OnLpmultiple orthogonal polynomials." Journal of Mathematical Analysis and Applications 407, no. 1 (November 2013): 147–56. http://dx.doi.org/10.1016/j.jmaa.2013.05.020.

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33

Livermore, Philip W. "Galerkin orthogonal polynomials." Journal of Computational Physics 229, no. 6 (March 2010): 2046–60. http://dx.doi.org/10.1016/j.jcp.2009.11.022.

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34

Draux, André, and Abdallah Maanaoui. "Vector orthogonal polynomials." Journal of Computational and Applied Mathematics 32, no. 1-2 (November 1990): 59–68. http://dx.doi.org/10.1016/0377-0427(90)90416-w.

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35

Pryzva, G. Y. "Kravchuk orthogonal polynomials." Ukrainian Mathematical Journal 44, no. 7 (July 1992): 792–800. http://dx.doi.org/10.1007/bf01056132.

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36

Celeghini, E., and M. A. del Olmo. "Coherent orthogonal polynomials." Annals of Physics 335 (August 2013): 78–85. http://dx.doi.org/10.1016/j.aop.2013.04.017.

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37

Fryant, A., A. Naftalevich, and M. K. Vemuri. "Orthogonal Homogeneous Polynomials." Advances in Applied Mathematics 22, no. 3 (April 1999): 371–79. http://dx.doi.org/10.1006/aama.1998.0637.

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38

Berriochoa, E., A. Cachafeiro, and J. M. Garcia-Amor. "A characterization of the four Chebyshev orthogonal families." International Journal of Mathematics and Mathematical Sciences 2005, no. 13 (2005): 2071–79. http://dx.doi.org/10.1155/ijmms.2005.2071.

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We obtain a property which characterizes the Chebyshev orthogonal polynomials of first, second, third, and fourth kind. Indeed, we prove that the four Chebyshev sequences are the unique classical orthogonal polynomial families such that their linear combinations, with fixed length and constant coefficients, can be orthogonal polynomial sequences.

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39

Berti, A. C., C. F. Bracciali, and A. Sri Ranga. "Orthogonal Polynomials Associated with Related Measures and Sobolev Orthogonal Polynomials." Numerical Algorithms 34, no. 2-4 (December 2003): 203–16. http://dx.doi.org/10.1023/b:numa.0000005363.32764.d3.

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40

Bracciali, C. F., A. P. da Silva, and A. Sri Ranga. "Szegő polynomials: some relations to L-orthogonal and orthogonal polynomials." Journal of Computational and Applied Mathematics 153, no. 1-2 (April 2003): 79–88. http://dx.doi.org/10.1016/s0377-0427(02)00605-2.

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41

Felix, H. M., A. Sri Ranga, and D. O. Veronese. "Kernel polynomials from L-orthogonal polynomials." Applied Numerical Mathematics 61, no. 5 (May 2011): 651–65. http://dx.doi.org/10.1016/j.apnum.2010.12.006.

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42

Ismail, Mourad E. H. "Relativistic orthogonal polynomials are Jacobi polynomials." Journal of Physics A: Mathematical and General 29, no. 12 (June 21, 1996): 3199–202. http://dx.doi.org/10.1088/0305-4470/29/12/023.

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43

Wang, Maw-Ling, Shwu-Yien Yang, and Rong-Yeu Chang. "Application of Generalized Orthogonal Polynomials to Parameter Estimation of Time-Invariant and Bilinear Systems." Journal of Dynamic Systems, Measurement, and Control 109, no. 1 (March 1, 1987): 7–13. http://dx.doi.org/10.1115/1.3143824.

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Generalized orthogonal polynomials (GOP) which can represent all types of orthogonal polynomials and nonorthogonal Taylor series are first introduced to estimate the parameters of a dynamic state equation. The integration operation matrix (IOP) and the differentiation operation matrix (DOP) of the GOP are derived. The key idea of deriving IOP and DOP of these polynomials is that any orthogonal polynomial can be expressed by a power series and vice versa. By employing the IOP approach to the identification of time invariant systems, algorithms for computation which are effective, simple and straightforward compared to other orthogonal polynomial approximations can be obtained. The main advantage of using the differentiation operation matrix is that the parameter estimation can be carried out starting at an arbitrary time of interest. In addition, the computational algorithm is even simpler than that of the integral operation matrix. Illustrative examples for using IOP and DOP approaches are given. Very satisfactory results are obtained.

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44

Ammar, Boukhemis, and Zerouki Ebtissem. "Classical2-orthogonal polynomials and differential equations." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–32. http://dx.doi.org/10.1155/ijmms/2006/12640.

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We construct the linear differential equations of third order satisfied by the classical2-orthogonal polynomials. We show that these differential equations have the following form:R4,n(x)Pn+3(3)(x)+R3,n(x)P″n+3(x)+R2,n(x)P′n+3(x)+R1,n(x)Pn+3(x)=0, where the coefficients{Rk,n(x)}k=1,4are polynomials whose degrees are, respectively, less than or equal to4,3,2, and1. We also show that the coefficientR4,n(x)can be written asR4,n(x)=F1,n(x)S3(x), whereS3(x)is a polynomial of degree less than or equal to3with coefficients independent ofnanddeg⁡(F1,n(x))≤1. We derive these equations in some cases and we also quote some classical2-orthogonal polynomials, which were the subject of a deep study.

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45

Zaghouani, Ali. "Some Basic 𝑑-Orthogonal Polynomial Sets." gmj 12, no. 4 (December 2005): 583–93. http://dx.doi.org/10.1515/gmj.2005.583.

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Abstract The purpose of this paper is to study the class of polynomial sets which are at the same time 𝑑-orthogonal and 𝑞-Appell. By a linear change of variable, the resulting set reduces to 𝑞-Al-Salam–Carlitz polynomials, for 𝑑 = 1. Various properties of the obtained polynomials are singled out: a generating function, a recurrence relation of order 𝑑 + 1. We also explicitly express a 𝑑-dimensional functional for which the d-orthogonality holds.

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46

Evans, Mogoi. "Orthogonal Polynomials and Operator Convergence in Hilbert Spaces: Norm-Attainability, Uniform Boundedness, and Compactness." Asian Research Journal of Mathematics 19, no. 10 (September 25, 2023): 227–34. http://dx.doi.org/10.9734/arjom/2023/v19i10744.

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This research paper investigates the convergence properties of operators constructed from orthogonal polynomials in the context of Hilbert spaces. The study establishes norm-attainability and explores the uniform boundedness of these operators, extending the analysis to include complex-valued orthogonal polynomials. Additionally, the paper uncovers connections between operator compactness and the convergence behaviors of orthogonal polynomial operators, revealing how sequences of these operators converge weakly to both identity and zero operators. These results advance our understanding of the intricate interplay betweenalgebraic and analytical properties in Hilbert spaces, contributing to fields such as functional analysis and approximation theory. The research sheds new light on the fundamental connections underlying the behavior of operators defined by orthogonal polynomials in diverse Hilbert space settings.

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47

Perev, K. "Orthogonal Polynomials Approximation and Balanced Truncation for a Lowpass Filter." Information Technologies and Control 11, no. 4 (December 1, 2013): 2–16. http://dx.doi.org/10.1515/itc-2015-0001.

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Abstract This paper considers the problem of orthogonal polynomial approximation based balanced truncation for a lowpass filter. The proposed method combines the system properties of balanced truncation, the computational effectiveness of proper orthogonal decomposition and the approximation capability of the orthogonal polynomials approximation. Orthogonal polynomials series expansion of the reachability and observability gramians is used in order to avoid solving large-scale Lyapunov equations and thus, significantly reducing the computational effort for obtaining the balancing transformation. The proposed method is applied for model reduction of a lowpass analog filter. Different sets of orthonormal functions are obtained from Legendre, Laguerre and Chebyshev orthogonal polynomials and the corresponding reduced order models are compared. The approximation precision is measured by the relative mean square error between the outputs of the full order model and the obtained reduced order models.

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48

Rebocho, Maria Das Neves. "Laguerre–Hahn orthogonal polynomials on the real line." Random Matrices: Theory and Applications 09, no. 01 (April 24, 2019): 2040001. http://dx.doi.org/10.1142/s2010326320400018.

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Abstract:

A survey is given on sequences of orthogonal polynomials related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coefficients — the so-called Laguerre–Hahn class. The main goal is to describe analytical aspects, focusing on differential equations for those orthogonal polynomials, difference and differential equations for the recurrence coefficients, and distributional equations for the corresponding linear functionals.

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49

QUESNE, C. "RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM." International Journal of Modern Physics A 26, no. 32 (December 30, 2011): 5337–47. http://dx.doi.org/10.1142/s0217751x11054942.

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A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to kth-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a (k-1)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invariance.

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50

Delvaux, Steven. "Average characteristic polynomials for multiple orthogonal polynomial ensembles." Journal of Approximation Theory 162, no. 5 (May 2010): 1033–67. http://dx.doi.org/10.1016/j.jat.2009.11.008.

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