60 research outputs found
Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm
The discrete-time Toda equation arises as a universal equation for the
relevant Hankel determinants associated with one-variable orthogonal
polynomials through the mechanism of adjacency, which amounts to the inclusion
of shifted weight functions in the orthogonality condition. In this paper we
extend this mechanism to a new class of two-variable orthogonal polynomials
where the variables are related via an elliptic curve. This leads to a `Higher
order Analogue of the Discrete-time Toda' (HADT) equation for the associated
Hankel determinants, together with its Lax pair, which is derived from the
relevant recurrence relations for the orthogonal polynomials. In a similar way
as the quotient-difference (QD) algorithm is related to the discrete-time Toda
equation, a novel quotient-quotient-difference (QQD) scheme is presented for
the HADT equation. We show that for both the HADT equation and the QQD scheme,
there exists well-posed -periodic initial value problems, for almost all
\s\in\Z^2. From the Lax-pairs we furthermore derive invariants for
corresponding reductions to dynamical mappings for some explicit examples.Comment: 38 page
Î-Coherent pairs of linear functionals and Markov-Bernstein inequalities
International audienceAll the linear functionals c0 and c1 associated to the Î-coherent pairs (c0,c1), have been given by Area et al. (2000). From these linear functionals c0 and c1 are introduced bilinear functionals aλ(p,q)=c0(pq)+λc1(ÎpÎq), âp,qâP, but only for the Î-coherent pairs (c0,c1) having a support Ω=]0,+â[. Five kinds of Î-coherent pairs given by Area et al. are concerned. They imply Charlier polynomials or Meixner polynomials, and they depend on a set of parameters. This paper is devoted to the study of the positivity of aλ(p,p) in order to obtain MarkovâBernstein inequalitiesc1((Îp)2)â€M2nc0(p2),âpâPn.The MarkovâBernstein constant Mnis equal to 1ÎŒ1,nâ where ÎŒ1,n is the smallest zero of a polynomial of degree n satisfying a three term recurrence relation. The five kinds of three term recurrence relation are obtained. The behavior of ÎŒ1,n is studied for a part of the variation of the parameters which characterize the Î-coherent pairs
On Asymptotics of the Sharp Constants of the MarkovâBernshtein Inequalities for the Sobolev Spaces
International audienc
Asymptotics of sharp constants of Markov-Bernstein inequalities in integral norm with Jacobi weight
International audienc
On Asymptotics of the Sharp Constants of the MarkovâBernshtein Inequalities for the Sobolev Spaces
Two spaces of generalized functions based on harmonic polynomials
Two spaces of generalized functions on the unit sphere Ωqâ1 â âq are introduced. Both types of generalized functions can be identified with suitable classes of harmonic functions. They are projective and inductive limits of Hilbert spaces. Several natural classes of continuous and continuously extendible operators are discussed: Multipliers, differentiations, harmonic contractions/expansions and harmonic shifts. The latter two classes of operators are "parametrized" by the full affine semigroup ân. AMS Classifications 46F05 46F10 31B05 20G0
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