Grupo de Análisis Matemático Aplicado (GAMA)
http://hdl.handle.net/10016/5857
2016-07-29T14:02:12ZSobolev orthogonal polynomials on product domains
http://hdl.handle.net/10016/23407
Sobolev orthogonal polynomials on product domains
Fenández, Lidia; Marcellán, Francisco; Pérez, Teresa E.; Pinar, Miguel A.; Xu, Yuan
Orthogonal polynomials on the product domain [a(1), b(1)] x [a(2), b(2)] with respect to the inner product < f, g >(s) = integral(b1)(a1) integral(b2)(a2) del f(x, y) center dot del g(x, y) w(1)(x)w(2)(y) dx dy +lambda f(c(1), c(2))g(c(1), c(2)) are constructed, where w(i) is a weight function on [a(i), b(i)] for i = 1, 2, lambda > 0, and (c(1), c(2)) is a fixed point. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions, which serve as primary examples.
2015-08-15T00:00:00Z(M, N) - Coherent pairs of linear functionals and Jacobi matrices
http://hdl.handle.net/10016/23406
(M, N) - Coherent pairs of linear functionals and Jacobi matrices
Marcellán, Francisco; Pinzón Cortés, Natalia Camila
A pair of regular linear functionals (U,V) in the linear space of polynomials with complex coefficients is said to be an (M,N) -coherent pair of order m if their corresponding sequences of monic orthogonal polynomials {Pn(x)}n⩾0 and {Qn(x)}n⩾0 satisfy a structure relation
∑i=0Mai,nP(m)n+m−i(x)=∑i=0Nbi,nQn−i(x),n⩾0,
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where M,N , and m are non-negative integers, {ai,n}n⩾0,0⩽i⩽M , and {bi,n}n⩾0,0⩽i⩽N , are sequences of complex numbers such that aM,n≠0 if n⩾M,bN,n≠0 if n⩾N , and ai,n=bi,n=0 if i>n . When m=1,(U,V) is called an (M,N) -coherent pair.
In this work, we give a matrix interpretation of (M,N) -coherent pairs of linear functionals. Indeed, an algebraic relation between the corresponding monic tridiagonal (Jacobi) matrices associated with such linear functionals is stated. As a particular situation, we analyze the case when one of the linear functionals is classical. Finally, the relation between the Jacobi matrices associated with (M,N) -coherent pairs of linear functionals of order m and the Hessenberg matrix associated with the multiplication operator in terms of the basis of monic polynomials orthogonal with respect to the Sobolev inner product defined by the pair (U,V) is deduced.
2014-04-01T00:00:00ZOn 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
2015-08-15T00:00:00ZMatrices totally positive relative to a tree, II
http://hdl.handle.net/10016/23404
Matrices totally positive relative to a tree, II
Costas Santos, Roberto; Johnson, Charles
If T is a labelled tree, a matrix A is totally positive relative to T , principal submatrices of A associated with deletion of pendent vertices of T are P-matrices, and A has positive determinant, then the smallest absolute eigenvalue of A is positive with multiplicity 1 and its eigenvector is signed according to T . This conclusion has been incorrectly conjectured under weaker hypotheses.
2016-09-15T00:00:00Z