Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 23 Jan 2020 18:07:06 GMT2020-01-23T18:07:06Z5051Hardy-type theorem for orthogonal functions with respect to their zeros. The Jacobi weight casehttp://hdl.handle.net/10316/4587Title: Hardy-type theorem for orthogonal functions with respect to their zeros. The Jacobi weight case
Authors: Abreu, L. D.; Marcellan, F.; Yakubovich, S. B.
Abstract: Motivated by the G.H. Hardy's 1939 results [G.H. Hardy, Notes on special systems of orthogonal functions II: On functions orthogonal with respect to their own zeros, J. London Math. Soc. 14 (1939) 37-44] on functions orthogonal with respect to their real zeros [lambda]n, , we will consider, under the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval (0,1), that is, the functions f(z)=z[nu]F(z), , where F is entire and when n[not equal to]m. Considering all possible functions on this class we obtain a new family of generalized Bessel functions including Bessel and hyperbessel functions as special cases.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/45872008-01-01T00:00:00ZVector interpretation of the matrix orthogonality on the real linehttp://hdl.handle.net/10316/13638Title: Vector interpretation of the matrix orthogonality on the real line
Authors: Branquinho, A.; Marcellán, F.; Mendes, A.
Abstract: In this paper we study sequences of vector orthogonal polynomials. The
vector orthogonality presented here provides a reinterpretation of what is known in
the literature as matrix orthogonality. These systems of orthogonal polynomials
satisfy three-term recurrence relations with matrix coefficients that do not obey
to any type of symmetry. In this sense the vectorial reinterpretation allows us to
study a non-symmetric case of the matrix orthogonality. We also prove that our
systems of polynomials are indeed orthonormal with respect to a complex measure
of orthogonality. Approximation problems of Hermite-Pad´e type are also discussed.
Finally, a Markov’s type theorem is presented.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/136382009-01-01T00:00:00ZRelative asymptotics for orthogonal matrix polynomialshttp://hdl.handle.net/10316/13707Title: Relative asymptotics for orthogonal matrix polynomials
Authors: Branquinho, A.; Marcellán, F.; Mendes, A.
Abstract: In this paper we study sequences of matrix polynomials that satisfy a
non-symmetric recurrence relation. To study this kind of sequences we use a vector
interpretation of the matrix orthogonality. In the context of these sequences of
matrix polynomials we introduce the concept of the generalized matrix Nevai class
and we give the ratio asymptotics between two consecutive polynomials belonging to
this class. We study the generalized matrix Chebyshev polynomials and we deduce
its explicit expression as well as we show some illustrative examples. The concept of
a Dirac delta functional is introduced. We show how the vector model that includes
a Dirac delta functional is a representation of a discrete Sobolev inner product. It
also allows to reinterpret such perturbations in the usual matrix Nevai class. Finally,
the relative asymptotics between a polynomial in the generalized matrix Nevai class
and a polynomial that is orthogonal to a modification of the corresponding matrix
measure by the addition of a Dirac delta functional is deduced.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10316/137072010-01-01T00:00:00ZCoherent pairs of linear functionals on the unit circlehttp://hdl.handle.net/10316/4581Title: Coherent pairs of linear functionals on the unit circle
Authors: Branquinho, A.; Moreno, A. Foulquié; Marcellán, F.; Rebocho, M. N.
Abstract: In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle: it is established that if ([mu]0,[mu]1) is a coherent pair of measures on the unit circle, then [mu]0 is a semi-classical measure. Moreover, we obtain that the linear functional associated with [mu]1 is a specific rational transformation of the linear functional corresponding to [mu]0. Some examples are given.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/45812008-01-01T00:00:00ZWKB Approximation and Krall-Type Orthogonal Polynomialshttp://hdl.handle.net/10316/7743Title: WKB Approximation and Krall-Type Orthogonal Polynomials
Authors: Álvarez-Nodarse, R.; Marcellán, F.; Petronilho, J.
Abstract: We give a unified approach to the Krall-type polynomials orthogonal withrespect to a positive measure consisting of an absolutely continuous one‘perturbed’ by the addition of one or more Dirac deltafunctions. Some examples studied by different authors are considered from aunique point of view. Also some properties of the Krall-type polynomials arestudied. The three-term recurrence relation is calculated explicitly, aswell as some asymptotic formulas. With special emphasis will be consideredthe second order differential equations that such polynomials satisfy. Theyallow us to obtain the central moments and the WKB approximation of thedistribution of zeros. Some examples coming from quadratic polynomialmappings and tridiagonal periodic matrices are also studied.
Thu, 01 Jan 1998 00:00:00 GMThttp://hdl.handle.net/10316/77431998-01-01T00:00:00Z