ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. The aim of the present paper is to generalize a result of Gábor Czédli. We estimate the maximum of the number of brick islands of ${\bf m_1\times m_2\times\cdots \times m_d}$, for any $d\geq3$ and for any fixed $d$-tuple $(m_1,m_2,\ldots,m_d)$ of positive integers.

AMS Subject Classification (1991): 05A16, 06D99

Keyword(s): lattice, distributive lattice, weakly independent subset, weak basis, brick island, number of islands

Received February 11, 2008, and in revised form February 18, 2008. (Registered under 6056/2009.)

Abstract. In 2003, G. Grätzer and E.$ $T. Schmidt introduced isoform lattices. A congruence relation on a lattice is {\it isoform}, if all the congruence classes are isomorphic sublattices. A lattice is {\it isoform}, if all of its congruences are isoform. They proved: {\it Every finite distributive lattice can be represented as the congruence lattice of a finite isoform lattice.} A much stronger result was proved by G. Grätzer, R.$ $W. Quackenbush, and E.$ $T. Schmidt in 2004: {\it Every finite lattice has a congruence-preserving extension to a finite isoform lattice.} They raised the problem whether this result can be extended to {\it congruence-finite} lattices, that is, to lattices with finitely many congruences. In this paper, we offer a positive solution of this problem, along with a somewhat easier proof of the original result.

AMS Subject Classification (1991): 06B10; 06B15

Keyword(s): congruence lattice, congruence-preserving extension, isoform, congruence-finite

Received July 20, 2007, and in revised form December 12, 2008. (Registered under 6057/2009.)

Abstract. We introduce {\it rectangular lattices}, a special type of planar semimodular lattices. We show that every finite distributive lattice can be represented as the congruence lattice of a ``small'' rectangular lattice, improving a 1998 result of G. Grätzer, H. Lakser, and E.$ $T. Schmidt.

AMS Subject Classification (1991): 06C10; 06B10

Keyword(s): semimodular lattice, planar, congruence, rectangular

Received October 15, 2007, and in revised form October 30, 2008. (Registered under 6058/2009.)

Abstract. A subset $X$ of a lattice $L$ with 0 is called {\it CDW-independent} if (1) it is CD-independent, i.e., for any $x,y\in X$, either $x\le y$ or $y\le x$ or $x\wedge y=0$ and (2) it is weakly independent, i.e., for any $n\in{\msbm N}$ and $x,y_1,\ldots, y_n\in X$ the inequality $x\le y_1\vee\cdots \vee y_n$ implies $x\le y_i$ for some $i$. A maximal CDW-independent subset is called a CDW-basis. With combinatorial examples and motivations in the background, the present paper points out that any two CDW-bases of a finite distributive lattice have the same number of elements. Moreover, if a lattice variety ${\cal V}$ contains a nondistributive lattice then there exists a finite lattice $L$ in ${\cal V}$ such that $L$ has CDW-bases $X$ and $Y$ with $|X|\not=|Y|$.

AMS Subject Classification (1991): 06D99

Keyword(s): lattice, distributivity, semimodularity, independent subset, CD-independent subset, weakly independent subset, CDW-independent subset, CDW-basis CD-basis

Received April 7, 2008, and in revised form November 10, 2008. (Registered under 6059/2009.)

Abstract. A common generalization of the Cayley theorem for monoids and that for (bounded) distributive lattices is presented.

AMS Subject Classification (1991): 08A05, 08A62

Keyword(s): Cayley theorem, monoid, (bounded) distributive lattice

Received January 21, 2008, and in revised form March 31, 2008. (Registered under 6060/2009.)

Abstract. Starting in 2001, Kemprasit and her students showed that certain semigroups of (linear) transformations defined on a set (or on a vector space) belong to $BQ$: that is, the class of semigroups $S$ in which every bi-ideal of $S$ is a quasi-ideal of $S$. Here, we unify that work and show that some of it can be derived from a result for abstract semigroups. We also extend their work, and that of Mendes-Gonçalves and Sullivan, to other examples of transformation semigroup.

AMS Subject Classification (1991): 20M20; 15A04

Keyword(s): quasi-ideal, bi-ideal, BQ, -semigroup, transformation semigroup

Received February 14, 2008, and in revised form October 17, 2008. (Registered under 6061/2009.)

Abstract. We consider a class of subsets of the general Sierpinski carpet which are characterized by insisting that the allowed digits in the expansion occur with prescribed group frequencies, determine their Hausdorff dimensions and give the necessary and sufficient conditions for their corresponding Hausdorff measures to be positive finite.

AMS Subject Classification (1991): 28A80, 28A78

Keyword(s): Hausdorff dimension, Sierpinski carpets, Hausdorff measure

Received July 4, 2007, and in revised form September 16, 2008. (Registered under 20/2007.)

Abstract. We establish the topological reflexivity of several spaces of analytic functions for which characterizations of their respective isometry groups are available. Namely, we consider the following spaces of analytic functions: the Novinger--Oberlin spaces consisting of those functions on the disk with the property that $f'\in H_p$ and also the more general Kolaski spaces; the Ida--Mochizuki spaces consisting of functions on the disk such that $$\sup_{0 < r < 1}\int_{\rm T}\log{(1 + |f(r\xi )|})^p d\sigma(\xi )< \infty $$ (with $p \geq1$, ${\rm T}$ is the unit circle and $d\sigma $ denotes Lebesgue measure); the subspace of the Nevanlinna class in several variables, known as the Smirnov Class, consisting of holomorphic functions $f$ on $X$ ($X$ the unit ball or the polydisk in $C^n$) so that $$\sup_{0\leq r< 1} \int_{\partial X} \log ^+(|f(rz)|) d\sigma(z) =\int_{\partial X} \log ^+(|f(z)|) d\sigma(z) < \infty.$$

AMS Subject Classification (1991): 30D55; 30D05

Keyword(s): isometries, local surjective isometries, algebraically reflexive Banach spaces, topologically reflexive Banach spaces

Received December 18, 2007, and in revised form December 15, 2008. (Registered under 6062/2009.)

Abstract. This paper provides a description of generalized bi-circular projections on Banach spaces of Lipschitz functions.

AMS Subject Classification (1991): 30D55; 30D05

Keyword(s): generalized bi-circular projections, surjective isometries, spaces of Lipshitz functions

Received February 20, 2008, and in revised form April 8, 2008. (Registered under 6063/2009.)

Abstract. In this paper we study the behavior of the difference equation $$ x_{n+1}=ax_{n}+{bx_{n}x_{n-2}\over cx_{n-1}+dx_{n-2}}, n=0,1,\ldots, $$ where the initial values $x_{-2}$, $x_{-1}$, $x_{0}$ are arbitrary positive real numbers and $a$, $b$, $c$, $d$ are positive constants. Also, we give the solution of some special cases of this equation.

AMS Subject Classification (1991): 39A10

Keyword(s): stability, boundedness, solution of difference equations

Received December 19, 2007, and in final form June 6, 2008. (Registered under 6064/2009.)

Abstract. In the present paper we are going to investigate Wilson's functional equation, where unknown functions are defined on a group and taking their values in an algebra. We express the solution of Wilson's equation in terms of absolutely convergent series of elements of a Banach algebra.

AMS Subject Classification (1991): 39B52

Keyword(s): trigonometric functions, Wilson's equation

Received May 19, 2008, and in revised form June 27, 2008. (Registered under 6065/2009.)

Abstract. The classical Weierstrass theorem states that any function continuous on a compact set $K\subset{\bf R}^d (d\ge1)$ can be uniformly approximated by algebraic polynomials. In this paper we study a possible extension of this celebrated result for approximation by {\it homogeneous} algebraic polynomials on {\it convex} surfaces ${K\subset\bf R}^d$ such that $K=-K$. Here we make a major progress in a previous conjecture proving that functions continuous on regular {\bf0}-symmetric convex surfaces can be approximated by a {\it pair} of homogeneous polynomials. Moreover, we settle completely the conjecture in $L_p$ metric when $1\le p< \infty $.

AMS Subject Classification (1991): 41A10, 41A63

Keyword(s): Jackson, uniform approximation, homogeneous polynomials, convex body

Received August 8, 2008, and in revised form November 17, 2008. (Registered under 6066/2009.)

Abstract. We study the differentiability properties of a function $f$ with absolutely convergent Fourier series and the smoothness property of the $r$th derivative $f^{(r)}$, where $r$ is a given natural number. We give best possible sufficient conditions in terms of the Fourier coefficients of $f$ to ensure that $f^{(r)}$ belongs either to one of the Lipschitz classes $\mathop{\rm Lip}(\alpha )$ and ${\rm lip}(\alpha )$ for some $0<\alpha < 1$, or to one of the Zygmund classes $\mathop{\rm Zyg}(1)$ and ${\rm zyg}(1)$. These sufficient conditions are also necessary in the cases when the Fourier coefficients $c_k$ of $f$ are real numbers such that either $k c_k\ge0$ for all $k$ or $c_k \ge0$ for all $k$.

AMS Subject Classification (1991): 42A32; 26A16, 26A24

Keyword(s): absolutely convergent Fourier series, Lipschitz classes and Zygmund classes of functions, formal differentiation of Fourier series

Received March 4, 2008. (Registered under 6067/2009.)

Abstract. Let $E$ be a subset of a discrete abelian group $\Gamma $ with dual group $G$. We say $E$ is $I_0(U)$ if every bounded function on $E$ is the restriction of the Fourier--Stieltjes transform of a discrete measure on $U$. We show that every $I_0(G)$ set is a finite union of $I_0(U)$ sets (the number is not independent of the open set $U$, but the dependancy is made clear); if $G$ is connected then $E$ is $I_0(U)$ for all open $U$. Related results are given.

AMS Subject Classification (1991): 42A55, 43A46; 43A05, 43A25

Keyword(s): associated sets, Bohr group, Fatou--Zygmund property, Hadamard sets, I_0, sets, Sidon sets

Received May 14, 2008, and in revised form July 18, 2008. (Registered under 6068/2009.)

Abstract. A general summability method, the so-called $\theta $-summability is considered for multi-dimensional Fourier series and Fourier transforms. Under some conditions on $\theta $ we will show that the restricted maximal operator of the $\theta $-means of a distribution is bounded from $H_p({\msbm T}^d)$ to $L_p({\msbm T}^d)$ for all $p_0< p\leq\infty $ and it is of weak type $(1,1)$, provided that the supremum in the maximal operator is taken over a cone-like set. The parameter $p_0< 1$ is depending on the dimension, the function $\theta $ and on the cone-like set. As a consequence we obtain a generalization of a well-known result due to Marcinkiewicz and Zygmund, namely, that the $d$-dimensional $\theta $-means of a function $f \in L_1({\msbm T}^d)$ converge a.e. to $f$ over the cone-like set. The same results are given for Fourier transforms, too. Some special cases of the $\theta $-summation are considered, such as the Cesàro, Fejér, Riesz, Riemann, Weierstrass, Picar, Bessel, Rogosinski and de La Vallée-Poussin summations.

AMS Subject Classification (1991): 42B08, 42A38, 42A24; 42B30

Keyword(s): Hardy spaces, p, -atom, Wiener algebra, \theta, -summation of Fourier series and Fourier transforms, restricted convergence, cone-like sets

Received March 14, 2008, and in revised form June 6, 2008. (Registered under 6069/2009.)

Abstract. A discrete version of the $\theta $-summability is introduced for higher dimensions. It is proved that the discrete $\theta $-means of a continuous function converge uniformly to the function. Moreover, the multi-dimensional Jackson polynomials converge uniformly to the continuous function.

AMS Subject Classification (1991): 42B08, 46E30, 42A38

Keyword(s): Wiener amalgam spaces, Herz spaces, \theta, discrete-summability, Fejér summability, Jackson polynomials, Hermite interpolation

Received October 14, 2008, and in revised form January 6, 2009. (Registered under 6070/2009.)

Abstract. Some characterizations of Sobolev spaces are discussed.

AMS Subject Classification (1991): 42B10, 42B35

Keyword(s): Schrödinger equation, initial value problems, oscillatory integrals, maximal estimates, Sobolev spaces

Received September 2, 2008, and in revised form December 16, 2008. (Registered under 6071/2009.)

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 ${\cal H}$ by $$(1)\qquad {\cal H} f(x) = \lim_{\epsilon,\eta\to 0}\int_{\epsilon\le |s| \le1} \int_{\eta\le |t|\le1 }f (x-P(s,t)) {ds dt\over st}. $$ We determine a necessary and sufficient condition on $P(s,t)$ so that the operator ${\cal H}$ is bounded on $L^p({\msbm R})$ for $1 < p < \infty $.

AMS Subject Classification (1991): 42B20

Keyword(s): Calderón--Zygmund kernel, multiplier, van der Corput's lemma

Received August 26, 2008, and in revised form October 20, 2008. (Registered under 6072/2009.)

Abstract. From the works of D.$ $V. Giang and F. Móricz (see [5]) and B.$ $I. Golubov (see [7]) it follows that the Hardy--Littlewood operator ${\cal B}(f)(x)=x^{-1}\int ^x_0f(t) dt$, $x\not=0$, is bounded on $BMO({\msbm R})$. We prove that ${\cal B}$ is also bounded on $VMO({\msbm R})$ and that the generalized Lipschitz classes $H^{\omega }_X({\msbm R})$ under additional conditions are invariant with respect to the operator ${\cal B}$. A direct approximation theorem for $VMO({\msbm R})$ is also obtained.

AMS Subject Classification (1991): 44A15, 47B38, 41A17

Keyword(s): Hardy--Littlewood operator, generalized Lipschitz classes, real Hardy space, functions of vanishing mean oscillation, direct approximation theorem

Received May 19, 2008, and in revised form October 29, 2008. (Registered under 6073/2009.)

Abstract. We investigate the link of cyclic behavior between a bounded operator and its Generalized Aluthge transforms. As an application, we characterize $\omega $-hyponormal operators for which the adjoint is hypercyclic or supercylic in terms of analytic spectral spaces. This extends a recent result of N. Feldman, V. G. Miller and T. L. Miller given for hyponormal operators.

AMS Subject Classification (1991): 47A10, 47A11, 47B20

Keyword(s): hypercyclic, supercyclic, \omega, -hyponormal, generalized Aluthge transforms

Received March 28, 2008, and in revised form September 17, 2008. (Registered under 6074/2009.)

Abstract. For a finite-dimensional operator $A$ with spectrum $\sigma(A)$, the following conditions on the Davis--Wielandt shell $DW(A)$ of $A$ are equivalent: (a) $A$ is normal. (b) $DW(A)$ is the convex hull of the set $\{(\lambda,|\lambda |^2): \lambda\in \sigma(A)\}.$ (c) $DW(A)$ is a polyhedron. These conditions are no longer equivalent for an infinite-dimensional operator $A$. In this note, a thorough analysis is given for the implication relations among these conditions. From the main result, one can deduce the equivalent conditions (a)--(c) for a finite-dimensional operator $A$, and show that the Davis--Wielandt shell cannot be used to detect normality for infinite-dimensional operators.

AMS Subject Classification (1991): 47A10, 47A12, 47B15

Keyword(s): Davis--Wielandt shell, numerical range, spectra, operator

Received January 15, 2008, and in revised form March 14, 2008. (Registered under 6075/2009.)

Abstract. Recently Gao--Yang showed the following result. Let $T$ be a $p$-hyponormal operator for $0< p \le1$. Then $$ (T^{n+1^*}T^{n+1})^{n+p\over n+1} \ge(T^{n^*}T^n)^{n+p\over n} \mbox{ and } (T^nT^{n^*})^{n+p\over n} \ge(T^{n+1}T^{n+1^*})^{n+p\over n+1} $$ hold for all positive integers $n$. Moreover, parallel results to invertible log-hyponormal operators have already been shown by Yamazaki, and also it was known that Yamazaki's result holds even for class $A$ operators. In this paper, as a parallel result to that of class $A$ operators, we shall show that the above inequalities hold under weaker conditions than $p$-hyponomality, that is, class $F(p,r,q)$ defined by Fujii--Nakamoto or class $wF(p,r,q)$ defined by Yang--Yuan under appropriate conditions of $p$, $r$ and $q$.

AMS Subject Classification (1991): 47B20, 47A63

Keyword(s): p, -hyponormal operators, log-hyponormal operators, A, classoperators, F(p, class, r, operators and class, q)wF(p, r, operators, q)

Received December 20, 2007. (Registered under 6076/2009.)

Abstract. Let $K$ be a convex body in ${\msbm E}^3$ with a $C^2$ smooth boundary. In this article, we investigate polytopes with at most $n$ edges circumscribed about $K$ or inscribed in $K$, which approximate $K$ best in the Hausdorff metric. The asymptotic behaviour of the distance, as a function of $n$, of such best approximating polytopes and $K$ is known, see [3] for an asymptotic formula. In this article, we prove that the typical faces of the best approximating circumscribed or inscribed polytopes in the Hausdorff metric with at most $n$ edges are asymptotically squares with respect to the second fundamental form of $\partial K$.

AMS Subject Classification (1991): 52A27, 52A50

Keyword(s): polytopal approximation, extremal problems, Hausdorff distance

Received March 6, 2008, and in revised form May 22, 2008. (Registered under 1/2008.)

Abstract. Multivariate normal distributions are described by a positive definite matrix and if their joint distribution is Gaussian as well then it can be represented by a block matrix. The aim of this note is to study Markov triplets by using the block matrix technique. A Markov triplet is characterized by the form of its block covariance matrix and by the form of the inverse of this matrix. A strong subadditivity of entropy is proved for a triplet and equality corresponds to the Markov property. The results are applied to multivariate stationary homogeneous Gaussian Markov chains.

AMS Subject Classification (1991): 54C70, 60J05; 40C05, 60G15

Keyword(s): normal distributions, Markov property, entropy, Schur complement, Hida--Cramér representation, Markov chain

Received March 15, 2008, and in revised form October 12, 2008. (Registered under 6077/2009.)

Abstract. In a largely overlooked paper of Murray Rosenblatt from more than thirty years ago, it was shown that if a strictly stationary Markov chain satisfies ``uniform ergodicity" as well as mixing (in the ergodic-theoretic sense), then it satisfies the (Rosenblatt) strong mixing condition. In this note, a strengthened version of that theorem will be presented. As a by-product, a couple of other, somewhat hidden, insights in that paper of Rosenblatt will be brought into sharper focus.

AMS Subject Classification (1991): 60G10, 60J05

Keyword(s): strong mixing, uniform ergodicity, Markov chain

Received July 30, 2008. (Registered under 6078/2009.)