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1
Content available remote

Tadek Pytlik in my memories

100%
Szwarc R.
Colloquium Mathematicum
|
2007
|
tom 108
|
nr 1
1-6
2
Content available remote

Nonnegative linearization of orthogonal polynomials

100%
Szwarc R.
Colloquium Mathematicum
|
1996
|
tom 69
|
nr 2
309-316
3
Content available remote

Norm estimates of discrete Schrödinger operators

100%
Szwarc R.
Colloquium Mathematicum
|
1998
|
tom 76
|
nr 1
153-160
EN
Harper's operator is defined on $\ell^2({\sym Z})$ by $$ H_\theta \xi(n) = \xi(n+1) + \xi(n-1) + 2\cos n\theta\, \xi(n), $$ where $\theta\! \in \![0,\pi]$. We show that the norm of $\|H_\theta\|$ is less than or equal to $2\sqrt{2}$ for $\pi/2 \le\theta\le \pi$. This solves a conjecture stated in [1]. A general formula for estimating the norm of self-adjoint tridiagonal infinite matrices is also derived.
4
Content available remote

Structure of geodesics in the Cayley graph of infinite Coxeter groups

100%
Szwarc R.
Colloquium Mathematicum
|
2003
|
tom 95
|
nr 1
79-90
EN
Let (W,S) be a Coxeter system such that no two generators in S commute. Assume that the Cayley graph of (W,S) does not contain adjacent hexagons. Then for any two vertices x and y in the Cayley graph of W and any number k ≤ d = dist(x,y) there are at most two vertices z such that dist(x,z) = k and dist(z,y) = d - k. Allowing adjacent hexagons, but assuming that no three hexagons can be adjacent to each other, we show that the number of such intermediate vertices at a given distance from x and y is at most 3. This means that the group W is hyperbolic in a sense stronger than that of Gromov.
5
Content available remote

The ratio and generating function of cogrowth coefficients of finitely generated groups

100%
Szwarc R.
Studia Mathematica
|
1998
|
tom 131
|
nr 1
89-94
EN
Let G be a group generated by r elements $g_1,…,g_r$. Among the reduced words in $g_1,…,g_r$ of length n some, say $γ_n$, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of $γ_{2n}$ has a limit, called the cogrowth exponent with respect to the generators $g_1,…,g_r$. We show by analytic methods that the numbers $γ_n$ vary regularly, i.e. the ratio $γ_{2n+2}/γ_{2n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated with the coefficients $γ_n$.
6
Content available remote

Kaczmarz algorithm in Hilbert space

64%
Haller R., Szwarc R.
Studia Mathematica
|
2005
|
tom 169
|
nr 2
123-132
EN
The aim of the Kaczmarz algorithm is to reconstruct an element in a Hilbert space from data given by inner products of this element with a given sequence of vectors. The main result characterizes sequences of vectors leading to reconstruction of any element in the space. This generalizes some results of Kwapień and Mycielski.
7
Content available remote

Kaczmarz algorithm with relaxation in Hilbert space

64%
Szwarc R., Świderski G.
Studia Mathematica
|
2013
|
tom 216
|
nr 3
237-243
EN
We study the relaxed Kaczmarz algorithm in Hilbert space. The connection with the non-relaxed algorithm is examined. In particular we give sufficient conditions when relaxation leads to the convergence of the algorithm independently of the relaxation coefficients.
8
Content available remote

Jacobi matrices on trees

64%
Kazun A., Szwarc R.
Colloquium Mathematicum
|
2010
|
tom 118
|
nr 2
465-497
EN
Symmetric Jacobi matrices on one sided homogeneous trees are studied. Essential selfadjointness of these matrices turns out to depend on the structure of the tree. If a tree has one end and infinitely many origin points the matrix is always essentially selfadjoint independently of the growth of its coefficients. In case a tree has one origin and infinitely many ends, the essential selfadjointness is equivalent to that of an ordinary Jacobi matrix obtained by restriction to the so called radial functions. For nonselfadjoint matrices the defect spaces are described in terms of the Poisson kernel associated with the boundary of the tree.
9
Content available remote

Weak type radial convolution operators on free groups

64%
Pytlik T., Szwarc R.
Studia Mathematica
|
2008
|
tom 186
|
nr 1
77-86
EN
Radial convolution operators on free groups with nonnegative kernel of weak type (2,2) and of restricted weak type (2,2) are characterized. Estimates of weak type (p,p) are obtained as well for 1 < p < 2.
10
Content available remote

Matrix coefficients of irreducible representations of free products of groups

44%
Szwarc R.
Studia Mathematica
|
1989
|
tom 94
|
nr 2
179-185
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