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1
Content available remote

Recurrence relations with periodic coefficients and Chebyshev polynomials

100%
Beckermann B., Gilewicz J., Leopold E.
Applicationes Mathematicae
|
1995-1996
|
tom 23
|
nr 3
319-323
EN
We show that polynomials defined by recurrence relations with periodic coefficients may be represented with the help of Chebyshev polynomials of the second kind.
2
Content available remote

A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms

80%
El Guennouni A.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 4
477-488
EN
The Lanczos method for solving systems of linear equations is implemented by using some recurrence relationships between polynomials of a family of formal orthogonal polynomials or between those of two adjacent families of formal orthogonal polynomials. A division by zero can occur in these relations, thus producing a breakdown in the algorithm which has to be stopped. In this paper, three strategies to avoid this drawback are discussed: the MRZ and its variants, the normalized and unnormalized BIORES algorithm and the composite step biconjugate algorithm. We prove that all these algorithms can be derived from a unified framework; in fact, we give a formalism for finding all the recurrence relationships used in these algorithms, which shows that the three strategies use the same techniques.
3
Content available remote

Orthogonal polynomials and the Lanczos method

80%
Brezinski C., Sadok H., Redivo Zaglia M.
Banach Center Publications
|
1994
|
tom 29
|
nr 1
19-33
EN
Lanczos method for solving a system of linear equations is well known. It is derived from a generalization of the method of moments and one of its main interests is that it provides the exact answer in at most n steps where n is the dimension of the system. Lanczos method can be implemented via several recursive algorithms known as Orthodir, Orthomin, Orthores, Biconjugate gradient,... In this paper, we show that all these procedures can be explained within the framework of formal orthogonal polynomials. This theory also provides a natural basis for curing breakdown and near-breakdown in these algorithms. The case of the conjugate gradient squared method can be treated similarly.
4
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Some counterexamples to subexponential growth of orthogonal polynomials

80%
Zygmunt M. J.
Studia Mathematica
|
1995
|
tom 116
|
nr 2
197-206
EN
We give examples of polynomials p(n) orthonormal with respect to a measure μ on ⨍ such that the sequence {p(n,x)} has exponential lower bound for some points x of supp μ. Moreover, the set of such points is dense in the support of μ.
5
Content available remote

On zeros of regular orthogonal polynomials on the unit circle

80%
Lázaro P., Marcellán F.
Annales Polonici Mathematici
|
1993
|
tom 58
|
nr 3
287-298
EN
A new approach to the study of zeros of orthogonal polynomials with respect to an Hermitian and regular linear functional is presented. Some results concerning zeros of kernels are given.
6
Content available remote

Limit points of eigenvalues of truncated unbounded tridiagonal operators

70%
Ifantis E. K., Kokologiannaki C. G., Petropoulou E.
Open Mathematics
|
2007
|
tom 5
|
nr 2
335-344
EN
Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e n}n=1∞, σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T N. We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.
7
Content available remote

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

61%
Dominici D.
Open Mathematics
|
2007
|
tom 5
|
nr 2
280-304
EN
We analyze the Charlier polynomials C n(χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.
8
Content available remote

On block recursions, Askey's sieved Jacobi polynomials and two related systems

61%
Aldana B. H., Charris J. A., Mora-Valbuena O.
Colloquium Mathematicum
|
1998
|
tom 78
|
nr 1
57-91
EN
Two systems of sieved Jacobi polynomials introduced by R. Askey are considered. Their orthogonality measures are determined via the theory of blocks of recurrence relations, circumventing any resort to properties of the Askey-Wilson polynomials. The connection with polynomial mappings is examined. Some naturally related systems are also dealt with and a simple procedure to compute their orthogonality measures is devised which seems to be applicable in many other instances.
9
Content available remote

Duality triads of higher rank: Further properties and some examples

61%
Schork M.
Open Mathematics
|
2006
|
tom 4
|
nr 3
507-524
EN
It is shown that duality triads of higher rank are closely related to orthogonal matrix polynomials on the real line. Furthermore, some examples of duality triads of higher rank are discussed. In particular, it is shown that the generalized Stirling numbers of rank r give rise to a duality triad of rank r.
10
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An extension of typically-real functions and associated orthogonal polynomials

61%
Naraniecka I., Szynal J., Tatarczak A.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
|
2011
|
tom 65
|
nr 2
EN
Two-parameters extension of the family of typically-real functions is studied. The definition is obtained by the Stjeltjes integral formula. The kernel function in this definition serves as a generating function for some family of orthogonal polynomials generalizing Chebyshev polynomials of the second kind. The results of this paper concern the exact region of local univalence, bounds for the radius of univalence, the coefficient problems within the considered family as well as the basic properties of obtained orthogonal polynomials.
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