DM - GAMA - Comunicaciones en Congresos y otros eventos
http://hdl.handle.net/10016/5859
2016-07-26T21:51:39ZOn linearly related sequences of difference derivatives of discrete orthogonal polynomials
http://hdl.handle.net/10016/23405
On linearly related sequences of difference derivatives of discrete orthogonal polynomials
Álvarez-Nodarse, Renato; Petronilho, José; Pinzón-Cortés, Natalia Camila; Sevinik-Adıgüzel, Rezan
Let ν be either ω∈C∖{0} or q∈C∖{0,1} , and let Dν be the corresponding difference operator defined in the usual way either by Dωp(x)=p(x+ω)−p(x)ω or Dqp(x)=p(qx)−p(x)(q−1)x . Let U and V be two moment regular linear functionals and let {Pn(x)}n≥0 and {Qn(x)}n≥0 be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete orthogonal polynomials involving the two OPS {Pn(x)}n≥0 and {Qn(x)}n≥0 assuming that their difference derivatives Dν of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as
∑Mi=0ai,nDmνPn+m−i(x)=∑Ni=0bi,nDkνQn+k−i(x),n≥0,
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where M,N,m,k∈N∪{0} , aM,n≠0 for n≥M , bN,n≠0 for n≥N , and ai,n=bi,n=0 for i>n . Under certain conditions, we prove that U and V are related by a rational factor (in the ν− distributional sense). Moreover, when m≠k then both U and V are Dν -semiclassical functionals. This leads us to the concept of (M,N) - Dν -coherent pair of order (m,k) extending to the discrete case several previous works. As an application we consider the OPS with respect to the following Sobolev-type inner product
⟨p(x),r(x)⟩λ,ν=⟨U,p(x)r(x)⟩+λ⟨V,(Dmνp)(x)(Dmνr)(x)⟩,λ>0,
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assuming that U and V (which, eventually, may be represented by discrete measures supported either on a uniform lattice if ν=ω , or on a q -lattice if ν=q ) constitute a (M,N) - Dν -coherent pair of order m (that is, an (M,N) - Dν -coherent pair of order (m,0) ), m∈N being fixed.
Proceedings of: OrthoQuad 2014. Puerto de la Cruz, Tenerife, Spain. January 20–24, 2014
2015-08-15T00:00:00ZZeros of Orthogonal Polynomials Generated by the Geronimus Perturbation of Measures
http://hdl.handle.net/10016/23375
Zeros of Orthogonal Polynomials Generated by the Geronimus Perturbation of Measures
Branquinho, Amílcar; Huertas Cejudo, Edmundo José; Rafaeli, Fernando R
This paper deals with monic orthogonal polynomial sequences
(MOPS in short) generated by a Geronimus canonical spectral transformation
of a positive Borel measure μ, i.e., (x−c)
−1dμ(x)+Nδ(x−c),
for some free parameter N ∈ IR+ and shift c. We analyze the behavior
of the corresponding MOPS. In particular, we obtain such a behavior
when the mass N tends to infinity as well as we characterize the precise
values of N such the smallest (respectively, the largest) zero of these
MOPS is located outside the support of the original measure μ. When
μ is semi-classical, we obtain the ladder operators and the second order
linear differential equation satisfied by the Geronimus perturbed MOPS,
and we also give an electrostatic interpretation of the zero distribution
in terms of a logarithmic potential interaction under the action of an
external field. We analyze such an equilibrium problem when the mass
point of the perturbation c is located outside the support of μ.
Proceedings of: 14th International Conference Computational Science and Its Applications (ICCSA 2014). Guimarães, Portugal, June 30 – July 3, 2014
2014-01-01T00:00:00ZEdge detection based on Krawtchouk polynomials
http://hdl.handle.net/10016/23357
Edge detection based on Krawtchouk polynomials
Rivero Castillo, Daniel; Pijeira, Héctor; Assunçao, Pedro
Discrete orthogonal polynomials are useful tools in digital image processing to extract visual object contours in different application contexts. This paper proposes an alternative method that extends beyond classic first-order differential operators, by using the properties of Krawtchouk orthogonal polynomials to achieve a first order differential operator. Therefore, smoothing of the image with a 2-D Gaussian filter is not necessary to regularize the ill-posed nature of differentiation. Experimentally, we provide simulation results which show that the proposed method achieves good performance in comparison with commonly used algorithms.
2015-08-15T00:00:00ZOn the convergence of type I Hermite-Padé approximants for rational perturbations of a Nikishin system
http://hdl.handle.net/10016/23353
On the convergence of type I Hermite-Padé approximants for rational perturbations of a Nikishin system
López Lagomasino, Guillermo; Medina Peralta, Sergio
We study the convergence of type I Hermite-Padé approximation for a class of meromorphic functions obtained by adding a vector of rational functions with real coefficients to a Nikishin system of functions.
2014-01-20T00:00:00Z