ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. In the present paper there are introduced and investigated the notions of fractal and semifractal lattice ordered groups or Riesz groups, respectively. The definitions are related to those which have been applied in the lattice theory. Besides, there is shown the existence of a proper class of Boolean algebras which are semifractal lattices but fail to be fractal lattices.

AMS Subject Classification (1991): 06F15, 06D35

Keyword(s): lattice ordered group ($\ell $-group, for short), fractal, semifractal, homogeneous Boolean algebra, Carathéodory functions

Received April 19, 2010, and in revised form November 23, 2010. (Registered under 30/2010.)

Abstract. We present two minimal clones containing 26 and 78 majority operations respectively, more than any other previously known example.

AMS Subject Classification (1991): 08A40

Keyword(s): clone, minimal clone, majority operation

Received March 29, 2010, and in revised form August 17, 2010. (Registered under 21/2010.)

Abstract. We give a short proof of a recent result of Hare, McKinnon and Sinclair on divisibility of the resultant of two polynomials whose roots are prime powers of a given monic polynomial. Our proof is based on Smyth's congruence involving powers of conjugate algebraic integers. An application to the Mahler measure of a polynomial is also given.

AMS Subject Classification (1991): 11C08, 11R04, 11R09

Keyword(s): polynomial, resultant, Mahler's measure

Received December 18, 2009. (Registered under 6439/2009.)

Abstract. The present paper continues work started by G. A. Muñoz-Fernández, Sz. Gy. Révész and J. B. Seoane-Sepúlveda [10] (degree 2 homogeneous polynomials, description of all extreme points) and L. Milev, N. Naidenov [8] (degree 2 algebraic polynomials, definite extreme points) by describing the indefinite extreme points of the unit ball of the space of degree 2 bivariate algebraic polynomials equipped with the maximum norm on the standard triangle of the plane. The main motivation for taking up this work is the hope that via the Krein--Milman theorem, this description will be useful in deriving the exact constants in certain inequalities, including the multivariate Bernstein inequality over general, non-symmetric convex bodies.

AMS Subject Classification (1991): 52A21, 26C05, 26B25

Keyword(s): convexity, extreme points, polynomials

Received January 8, 2010, and in final form May 18, 2010. (Registered under 3/2010.)

Abstract. Let $\kappa $ be a positive integer. A sequence $(f_n)$ of generalized Nevanlinna functions of the class ${\bf N}_\kappa $, which converges locally uniformly on some nonempty open subset of the complex plane to a function $f$, need not converge on any larger set, and $f$ can belong to any class $\bf N_{\kappa '}$ with $0\le\kappa '\le\kappa $. In this note we show that if it is a priori known that $f$ belongs to the same class ${\bf N}_\kappa $ then the sequence $(f_n)$ converges locally uniformly on the set $({\msbm C}\setminus{\msbm R})\cap{\rm hol}f$, and the sets of poles or generalized poles of nonpositive type of $f_n$ converge to the set of poles or generalized poles of nonpositive type of $f$. Moreover, a compactness result for generalized Nevanlinna functions is proved.

AMS Subject Classification (1991): 30E20, 30C15, 46C20, 46G99

Keyword(s): generalized Nevanlinna functions, rational functions, locally uniform convergence

Received February 25, 2011, and in revised form July 8, 2011. (Registered under 12/2011.)

Abstract. We give a new proof of the fact that any meromorphic solution of a second order Briot--Bouquet differential equation in the whole plane is a degenerate or non-degenerate elliptic function. Our argument does not depend on the first order case.

AMS Subject Classification (1991): 34A20, 30D05

Keyword(s): Briot--Bouquet differential equations

Received April 28, 2009, and in revised form August 17, 2010. (Registered under 59/2009.)

Abstract. In this note we return to some problems of general orthogonal series examined most intensively at the fifties. We show that the monotonicity conditions in four fundamental theorems of K. Tandori can be replaced by weaker assumptions.

AMS Subject Classification (1991): 40A30, 40G05

Keyword(s): orthogonal series, convergence, summability

Received March 2, 2010. (Registered under 16/2010.)

Abstract. The convergence of Cesàro means of negative order of double trigonometric Fourier series of functions of bounded partial $\Lambda $-variation is investigated. The sufficient and neccessary conditions on the sequence $\Lambda =\{\lambda_{n}\} $ are found for the convergence of Cesàro means of Fourier series of functions of bounded partial $\Lambda $-variation.

AMS Subject Classification (1991): 42B08

Keyword(s): Fourier series, $\Lambda $-variation, generalized variation, Cesàro means

Received March 12, 2010, and in revised form May 17, 2010. (Registered under 19/2010.)

Abstract. We prove the following theorem: {\it Suppose that $E\subset[0,2\pi )^2$ is any Lebesgue measurable set, $\mu_{2}E >0,$ and $\phi(u)$ is a nonnegative, continuous and nondecreasing function on $[0,\infty )$ such that $u\phi(u)$ is a convex function on $[0,\infty )$ and $ \phi(u) = o(\ln u), u \to\infty. $ Then there exists a function $g \in L_1([0,2\pi )^2)$ such that $ \int_{[0,2\pi )^2} | g(x,y) |\phi(| g(x,y) |)dx dy < \infty_{\strut }^{\strut } $ and the sequence of the strong logarithmic means by squares of the double trigonometric Fourier series of $g$, that is, the sequence $ \left\{{1\over\ln N}\sum_{k=1}^N {| S_{k,k}(g;x,y) - g (x,y)| \over k}, N=2,3,\ldots\right \} _{\strut }^{\strut } $ is not bounded in measure on $E$.}

AMS Subject Classification (1991): 42C15, 42C10

Keyword(s): double Fourier series, strong logarithmic means, bounded in measure

Received April 7, 2010. (Registered under 23/2010.)

Abstract. We consider the commutator operator $[B, \sigma(x,D)]$ of the multiplication operator by a function $B$ and a pseudo-differential operator $\sigma(x,D)$, and prove that $[B, \sigma(x,D)]$ is bounded on the local Hardy spaces $h^p({\msbm R}^n)$. We also show that our result is optimal.

AMS Subject Classification (1991): 42B20

Keyword(s): pseudo-differential operator, commutator, Hardy space, local Hardy space

Received January 13, 2009, and in revised form April 11, 2011. (Registered under 7/2009.)

Abstract. Let $P(s,t)$ denote a real-valued polynomial of real variables $s$ and $t$. For $f \in{\cal S}$ (i.e., a Schwartz class function), define the operator $T$ by (1) $ Tf(x,y) = \lim_{\epsilon,\eta\to 0}\int_{\epsilon\le |s| \le1} \int_{\eta\le |t|\le1 }f (x-s, y-P(s,t))_{\strut }^{\strut } {ds dt\over st}. $ We determine a necessary and sufficient condition on $P(s,t)$ so that the operator $T$ is bounded on $L^p({\msbm R}^2)$ for $1 < p < \infty $.

AMS Subject Classification (1991): 42B20

Keyword(s): Newton diagram, multiplier, van der Corput's lemma

Received January 20, 2010, and in revised form September 9, 2010. (Registered under 4/2010.)

Abstract. We investigate some properties of the class of semi-compact operators on Banach lattices and we give some interesting consequences. In particular we are interested in Banach lattices $E$ and $F$ such that (a) every order bounded operator $T\colon E\rightarrow F$ possessing a semi-compact adjoint is semi-compact as well, (b) every order bounded semi-compact operator $T\colon E\rightarrow F$ has a semi-compact adjoint.

AMS Subject Classification (1991): 46A40, 46B40, 46B42

Keyword(s): semi-compact operator, order continuous norm, discrete Banach lattice

Received March 5, 2010, and in revised form May 10, 2010. (Registered under 18/2010.)

Abstract. Various operator theoretic properties of composition operators with linear fractional symbol acting on the Dirichlet space of the unit ball are discussed. Furthermore, we use Calderón's complex interpolation to investigate the spectrum of composition operators with automorphic symbol acting on the analytic Besov spaces of the ball and on the weighted Dirichlet spaces of the ball, which include the Dirichlet, Arveson, Hardy and Bergman spaces.

AMS Subject Classification (1991): 47B33, 46B70

Keyword(s): composition operator, Dirichlet space, Besov space, spectrum, complex interpolation

Received December 1, 2009. (Registered under 6408/2009.)

Abstract. Let $\Pi_{\kappa }$ be a Pontryagin space with decomposition $\Pi_{\kappa }=\Pi_{+}(+)\Pi_{-}$, $\kappa =\dim\Pi _{+}< \infty $. Let $U$ be a unitary operator in $\Pi_{\kappa }$ and let $U= \left[{A B\atop C D}\right ]$ be its matrix representation that corresponds to the given decomposition of $\Pi_{\kappa }$. In this note operators $A$, $B$, $C$, and $D$ are given in terms of isometric operators and orthogonal projections in a way that those expressions are necessary and sufficient conditions for the operator $U$ to be unitary. The results are more specific and intuitive than the results from the last chapter of [2]. The obtained representation of $U$ is applied to study operator $T$ that has $\kappa $-dimensional positive invariant subspace $J_{+}$ and allows a J-polar decomposition. The radius of the spectrum $\sigma(T\mid J_{+}) $ is estimated.

AMS Subject Classification (1991): 47B50, 46C20

Keyword(s): Pontryagin space, unitary operator in $\Pi_{\kappa }$, polar decomposition

Received April 16, 2010, and in revised form May 6, 2011. (Registered under 29/2010.)

Abstract. The paper characterizes the continuity and compactness of the weighted composition operators $W_{(u,T)}$ on Orlicz--Lorentz spaces $L_{\varphi,w}$.

AMS Subject Classification (1991): 47B38, 46E30

Keyword(s): distribution function, Lorentz space, Orlicz space, Orlicz--Lorentz space, compact operator, weighted composition operator

Received December 11, 2009, and in revised form March 25, 2011. (Registered under 6415/2009.)

Abstract. We characterize those subpositive operators for which their Krein--von Neumann extension has closed range, moreover we construct their Moore--Penrose inverse. Our treatment follows as a tool the factorization approach to the extension theory of positive operators. As addition we give a short proof of Dixmier's theorem that a bounded positive operator $A$ and its square root $A^{1/2}$ have the same range if and only if $A$ has closed range and of Banach's closed range theorem for Hilbert space operators.

AMS Subject Classification (1991): 47A20, 47B65, 47A05

Keyword(s): characterization, positive operator, closed range, Krein--von Neumann extension, Moore--Penrose inverse

Received December 23, 2009, and in revised form February 1, 2011. (Registered under 6468/2009.)

Abstract. In [9] a question is raised: if a power bounded operator is quasisimilar to a singular unitary operator, is it similar to this unitary operator? For polynomially bounded operators, a positive answer to this question is known [1], [13]. In this paper a positive answer is given in some particular cases, but in general an answer rests unknown.

AMS Subject Classification (1991): 47A05, 47B99, 47B15

Keyword(s): power bounded operator, singular unitary operator, similarity, quasisimilarity, quasiaffine transform

Received April 8, 2010, and in revised form October 12, 2010. (Registered under 24/2010.)

Abstract. It is proved that every semigroup of contractions with parameter on ${\msbm Q}_{+} \times{\msbm Q}_{+}$ or ${\msbm Q}_{+} \times{\msbm N}$ has a unitary dilation. The dilation result about ${\msbm Q}_{+} \times{\msbm Q}_{+}$ is used to obtain a new proof of the Slociński dilation theorem, which says that every strongly continuous semigroup of contractions, with parameter on ${\msbm R}_{+} \times{\msbm R}_{+}$, has a strongly continuous unitary dilation. The result about ${\msbm Q}_{+} \times{\msbm N}$ is used to obtain a new proof of the continuous version of the commutant lifting theorem.

AMS Subject Classification (1991): 47A20, 47D03

Keyword(s): unitary dilation, semigroup of contractions, commutant lifting

Received November 29, 2009, and in revised form April 13, 2010. (Registered under 6233/2009.)

Abstract. We provide a characterization of compact weighted composition operators on spaces of vector-valued Lipschitz functions. We also give estimates of the essential norm of composition operators on these spaces.

AMS Subject Classification (1991): 47B33, 47B37

Keyword(s): weighted composition operators, essential norm

Received December 2, 2009. (Registered under 6412/2009.)

Abstract. We introduce the notion of an ${\msbm R}$-group of which the classical groups ${\msbm R}$, ${\msbm Z}$ and ${\msbm R}_{+}^{\ast }$ are typical examples, and we study flows $( X,{\cal H}) $, where $X$ is a locally compact space and ${\cal H}$ is a continuous ${\msbm R}$-group action on $X$ with the further property that any compact set is \hbox{\it absorbed }(in the ordinary meaning in use in the theory of topological vector spaces) by any neighbourhood of some characteristic point in $X$ called the center of ${\cal H}$. The case where $X$ is a locally compact abelian group is also considered. We are particularly interested in discussing the asymptotic properties of ${\cal H}$, which is made possible by proving a deep theorem about the existence of nontrivial ${\cal H}$-homogeneous positive measures on $X$. Also, a close connection with homogenization theory is pointed out. It appears that the present paper lays the foundation of the mathematical framework that is needed to undertake a systematic study of homogenization problems on manifolds, Lie groups included.

AMS Subject Classification (1991): 37B05, 43A07, 46J10, 28A25, 28A50, 26E60, 54D45

Keyword(s): locally compact space, group actions

Received May 1, 2010, and in revised form February 8, 2011. (Registered under 36/2010.)

Abstract. For $k\geq7$, we determine the minimal area of a compact hyperbolic surface, and an oriented compact hyperbolic surface that can be tiled by embedded regular triangles of angle $2\pi /k$. Based on this, all the cases of equality in László Fejes Tóth's triangle bound for hyperbolic surfaces are described.

AMS Subject Classification (1991): 51M10, 52C22

Keyword(s): tiling of hyperbolic surfaces

Received April 14, 2010, and in revised form January 11, 2011. (Registered under 28/2010.)

Abstract. We investigate the inhomogeneous Galton--Watson processes with immigration, where $\rho_n$, the offspring means in the $n^{\rm th}$ generation, tends to $1$. We show that if the second derivatives of the offspring generating functions go to $0$ rapidly enough, then the asymptotics are the same as in the INAR(1) case, treated in [4]. We also determine the limit if this assumption does not hold showing the optimality of the conditions.

AMS Subject Classification (1991): 60J80

Keyword(s): nearly critical Galton--Watson process, immigration, compound Poisson distribution, negative binomial distribution

Received October 20, 2010, and in revised form July 10, 2011. (Registered under 72/2010.)

Abstract. Durbin's estimated empirical process is a widely used tool to testing goodness of fit for parametric distribution families. In general, statistical methods based on the process are not distribution free and the critical values can not always be calculated in a theoretical way. One can avoid these difficulties by applying the parametric or the non-parametric bootstrap procedure. Although the parametric bootstrapped estimated empirical process is well investigated, only a few papers dealt with the non-parametric version. Recently, Babu and Rao pointed out that in the latter case a bias correction is needed, and they proved the weak convergence of the bootstrapped process in continuous distribution families. Our paper presents a weak approximation theorem for the non-parametric bootstrapped estimated empirical process using similar conditions under which Durbin's non-bootstrapped process converges. The result covers the most important continuous and discrete distribution families. Simulation studies in the Poisson and the normal distribution are also reported.

AMS Subject Classification (1991): 62E20, 62F40, 62G30

Keyword(s): bootstrap, parametric estimation, empirical process, approximation, convergence in distribution

Received December 30, 2010, and in revised form April 7, 2011. (Registered under 90/2010.)