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- Waldemar W. Koczkodaj, Ryszard Szwarc
- Fundam. Inform.
- 2014

This study examines the notion of inconsistency in pairwise comparisons for providing an axiomatization for it. It also proposes two inconsistency indicators for pairwise comparisons. The primary motivation for the inconsistency reduction is expressed by a computer industry concept " garbage in, garbage out ". The quality of the output depends on the… (More)

- Ryszard Szwarc
- 2008

The orthogonal polynomials p n satisfy Turán's inequality if p 2 n (x)− p n−1 (x)p n+1 (x) ≥ 0 for n ≥ 1 and for all x in the interval of or-thogonality. We give general criteria for orthogonal polynomials to satisfy Turán's inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the… (More)

- Christian Berg, Ryszard Szwarc
- Journal of Approximation Theory
- 2009

Let µ denote a symmetric probability measure on [−1, 1] and let (p n) be the corresponding orthogonal polynomials normalized such that p n (1) = 1. We prove that the normalized Turán determinant ∆ n (x)/(1 − x 2), where ∆ n = p 2 n − p n−1 p n+1 , is a Turán determinant of order n − 1 for orthogonal polynomials with respect to (1 − x 2)dµ(x). We use this to… (More)

We give a sufficient condition for strictly positive def-initeness in R d. The result is based on the question how sparse subsets of R d can be to guarantee linear independence of the ex-ponentials. Interpolation by positive definite functions has become a widely used technique in approximation theory and spatial statistics. The basic model is defined as… (More)

- RYSZARD SZWARC
- 2005

We prove that any tight frame in Hilbert space can be obtained by the Kaczmarz algorithm. An explicit way of constructing this correspondence is given. The uniqueness of the correspondence is determined.

Let H N = (s n+m), n, m ≤ N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue λ N of H N. It is proved that λ N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of λ N can be arbitrarily slow… (More)

- RYSZARD SZWARC
- 2009

New criteria for nonnegativity of connection coefficients between to systems of orthogonal polynomials are given. The results apply to classical orthogonal polynomials.

Let S ⊂ R denote a compact set with infinite cardinality and C(S) the set of real continuous functions on S. We investigate the problem of polynomial and orthogonal polynomial bases of C(S). In case of S = {s0, s1, s2,. . .} ∪ {σ}, where (s k) ∞ k=0 is a monotone sequence with σ = lim k→∞ s k , we give a sufficient and necessary condition for the existence… (More)

- Ryszard Szwarc
- 2004

Spectral properties of unbounded symmetric Jacobi matrices are studied. Under mild assumptions on the coefficients absolute continuity of spectral measure is proved. Only operator theoretic proofs are provided. Some open problems of Ifantis are solved.

- RYSZARD SZWARC
- 2010

Let G be a group generated by g\,...,gr ■ There are exactly 2r(2r — l)"_l reduced words in g\,...,gr of length n. Part of them, say y " represents identity element of G. Let y = lim sup y ". We give a short proof of the theorem of Grigorchuk and Cohen which states that G is amenable if and only if y = 2r-12. Moreover we derive some new properties of the… (More)