André DRAUX | INSA ROUEN (original) (raw)

Papers by André DRAUX

arXiv (Cornell University), May 11, 2016

All the coherent pairs of measures associated to linear functionals c0 and c1, introduced by Iser... more All the coherent pairs of measures associated to linear functionals c0 and c1, introduced by Iserles et al in 1991, have been given by Meijer in 1997. There exist seven kinds of coherent pairs. All these cases are explored in order to give three term recurrence relations satisfied by polynomials. The smallest zero µ1,n of each of them of degree n has a link with the Markov-Bernstein constant Mn appearing in the following Markov-Bernstein inequalities: c1((p ′) 2) ≤ M 2 n c0(p 2), ∀p ∈ Pn, where Mn = 1 √ µ 1,n. The seven kinds of three term recurrence relations are given. In the case where c0 = e −x dx + δ(0) and c1 = e −x dx, explicit upper and lower bounds are given for µ1,n, and the asymptotic behavior of the corresponding Markov-Bernstein constant is stated. Except in a part of one case, limn→∞ µ1,n = 0 is proved in all the cases.

Numerical Algorithms, 1997

Numerical Algorithms, 2002

Integral Transforms and Special Functions, 2016

ABSTRACT The sequences of quasi-orthogonal polynomials of order r are defined for non-quasi-defin... more ABSTRACT The sequences of quasi-orthogonal polynomials of order r are defined for non-quasi-definite moment functionals. Properties concerning the existence of such sequences, and relations between a quasi-orthogonal polynomial of order r and a set of orthogonal polynomials are proved. Two determinantal expressions of quasi-orthogonal polynomials of order r are given. At last it is proved that three consecutive polynomials of a sequence of quasi-orthogonal polynomials of order r satisfy a three term recurrence relation.

Lecture Notes in Control and Information Sciences

Without Abstract

Numerical Algorithms, 2008

The Markov–Bernstein inequalities for the Jacobi measure remained to be studied in detail. Indeed... more The Markov–Bernstein inequalities for the Jacobi measure remained to be studied in detail. Indeed the tools used for obtaining lower and upper bounds of the constant which appear in these inequalities, did not work, since it is linked with the smallest eigenvalue of a five diagonal positive definite symmetric matrix. The aim of this paper is to generalize the qd

Applied Numerical Mathematics, 1991

In this article properties of automatic selection of sequence transformations are proved for comp... more In this article properties of automatic selection of sequence transformations are proved for composite sequence transformations as well as properties of convergence acceleration. The use of a delaying transformation is pointed out to obtain convergence acceleration.

Different types of polynomial inequalities have been studied since more one century. The first ty... more Different types of polynomial inequalities have been studied since more one century. The first type is the so-called Markov-Bernstein inequalities (A. A. Markov in 1889 [5]). More recently other inequalities involving the Lnorm for the Hermite measure were given (see Bojanov and Varma [2], Alves and Dimitrov [1]). Inequalities given in the previous papers are particular cases of the more general ones which can be obtained by using the variational method. The basis of our study can be found in the papers of Draux and Elhami ([3], [4]). In this talk we will present the Landau-Kolmogorv inequalities obtained in the case of Hermite and closely connected measures.

Proceedings of the American Mathematical Society, 2015

The classical A. Markov inequality establishes a relation between the maximum modulus or the L ∞ ... more The classical A. Markov inequality establishes a relation between the maximum modulus or the L ∞ ([−1, 1]) norm of a polynomial Qn and of its derivative: Q ′ n Mnn 2 Qn , where the constant Mn = 1 is sharp. The limiting behavior of the sharp constants Mn for this inequality, considered in the space L 2 [−1, 1], w (α,β) with respect to the classical Jacobi weight w (α,β) (x) := (1 − x) α (x + 1) β , is studied. We prove that, under the condition |α − β| < 4, the limit is limn→∞ Mn = 1/(2jν) where jν is the smallest zero of the Bessel function Jν (x) and 2ν = min(α, β) − 1.

Annali di Matematica Pura ed Applicata, 1991

SummaryTwo-point Padé-type approximants are introduced in the case of a non-commutative algebra o... more SummaryTwo-point Padé-type approximants are introduced in the case of a non-commutative algebra on a commutative field. Algebraic properties are given and a study of the error of approximation is done. From the relation of the error and some additional properties, two-point Padé approximants are found. Algebraic properties and recurrence relations are proved. The means to compute these approximants in following any way in the table of the approximants are given. The mixed table is introduced, as well as the normality and some results of convergence of two-point Padé-type and Padé approximants.

Applied Numerical Mathematics, 2010

The block qd algorithm is studied in order to obtain some properties about the asymptotic behavio... more The block qd algorithm is studied in order to obtain some properties about the asymptotic behavior of some eigenvalues of a block tridiagonal positive definite symmetric matrix. We prove that the eigenvalues of the first block on the block diagonal of the decomposition given by the block qd algorithm at the different stages of this algorithm constitute strictly increasing sequences

Applied Numerical Mathematics, 2011

Gegenbauer polynomials Generalized Gegenbauer polynomials Quasi-orthogonal polynomials Eigenvalue... more Gegenbauer polynomials Generalized Gegenbauer polynomials Quasi-orthogonal polynomials Eigenvalue problem Generalized eigenvalue problem qd Algorithm The Markov-Bernstein inequalities for generalized Gegenbauer weight are studied. A special basis of the vector space P n of real polynomials in one variable of degree at most equal to n is proposed. It is produced by quasi-orthogonal polynomials with respect to this generalized Gegenbauer measure. Thanks to this basis the problem to find the Markov-Bernstein constant is separated in two eigenvalue problems. The first has a classical form and we are able to give lower and upper bounds of the Markov-Bernstein constant by using the Newton method and the classical qd algorithm applied to a sequence of orthogonal polynomials. The second is a generalized eigenvalue problem with a five diagonal matrix and a tridiagonal matrix. A lower bound is obtained by using the Newton method applied to the six term recurrence relation produced by the expansion of the characteristic determinant. The asymptotic behavior of an upper bound is studied. Finally, the asymptotic behavior of the Markov-Bernstein constant is O(n 2) in both cases.

This paper is devoted to Landau-Kolmogorov type inequalities in several variables in L norm for J... more This paper is devoted to Landau-Kolmogorov type inequalities in several variables in L norm for Jacobi measures. These measures are chosen in such a way that the partial derivatives of the Jacobi orthogonal polynomials are also orthogonal. These orthogonal polynomials in several variables are built by tensor product of the orthogonal polynomials in one variable. These inequalities are obtained by using a variational method and they involve the square norms of a polynomial p and at most those of the partial derivatives of order 2 of p.

Lobachevskii Journal of Mathematics

The Sobolev spaces with continuous and discrete coherent pairs of weights are considered. The pos... more The Sobolev spaces with continuous and discrete coherent pairs of weights are considered. The positivity of the inner product is equivalent to the Markov–Bernstein inequality for the weighted integral norm. Asymptotics of the sharp constants for these inequalities, when the degree of polynomials goes to infinity, are obtained.