Generalized qd algorithm and Markov–Bernstein inequalities for Jacobi weight

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...

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Vydáno v:Numerical algorithms Ročník 51; číslo 4; s. 429 - 447
Hlavní autoři: Draux, André, Moalla, Borhane, Sadik, Mohamed
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.08.2009
Springer Nature B.V
Springer Verlag
Témata:
Algebra
Algorithms
Computer Science
Constants
Determinants
Eigenvalues
Inequalities
Mathematical analysis
Mathematics
Matrices (mathematics)
Numeric Computing
Numerical Analysis
Oroginal Paper
Symmetry
Theory of Computation
Upper bounds
Jacobi polynomials
Gegenbauer polynomials
33C45
26C10
Orthogonal polynomials
LR algorithm
Markov–Bernstein inequalities
qd algorithm
42C05
26D05
ISSN:1017-1398, 1572-9265
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Popis
Shrnutí: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 algorithm for positive definite symmetric band matrices and to give the mean to expand the determinant of a five diagonal symmetric matrix. After that these new tools are applied to the problem to produce effective lower and upper bounds of the Markov–Bernstein constant in the Jacobi case. In the last part we com pare, in the particular case of the Gegenbauer measure, the lower and upper bounds which can be deduced from this paper, with those given in Draux and Elhami (Comput J Appl Math 106:203–243, 1999 ) and Draux (Numer Algor 24:31–58, 2000 ).
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-008-9241-4