ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. For integers $m_1,\ldots,m_d>0$ and a cuboid $M=[0,m_1]\times\cdots \times[0,m_d]\subset{\msbm R}^d$, a set $H$ of closed bricks in $M$ is a system of brick islands if, for each pair of bricks in $H$, one contains the other or they are disjoint. Such a system is maximal if it cannot be extended to a larger system of brick islands. We show that the minimum size of a maximal system of brick islands in $M$ is $\sum_{i=1}^d m_i - (d-1)$. Also, a system of cubic islands is a system of brick islands for which all the bricks are cubes. We show that the minimum size of a maximal system of cubic islands in a cube $C=[m]^d$ is $m$.

AMS Subject Classification (1991): 05A05

Keyword(s): brick islands

Received December 17, 2010, and in revised form April 19, 2012. (Registered under 89/2010.)

Abstract. Directoids are groupoids defined on every upward directed poset. They fully characterize these posets. Hence, in order to study conguences on directed posets, we can convert the poset into a directoid and study congruences on it. The paper is devoted to several characterizations of congruences on directoids, on directoids with an antitone involution, on directoids with sectionally antitone involutions and on double directoids.

AMS Subject Classification (1991): 06A06, 06B10, 08A30

Keyword(s): directoid, poset, congruence relation, filter, antitone involution, double directoid

Received October 11, 2011, and in revised form April 14, 2012. (Registered under 53/2011.)

Abstract. The aim of the present paper is the estimation of the $d$th moment of additive functions in canonical number systems. These number systems are generalizations of the decimal number system to arbitrary polynomials having integer coefficients. We call a function additive (with respect to a number system) if it only acts on the digits of an expansion. The sum-of-digits function, as a special additive function, has been analyzed in the case of $q$-adic number systems by Delange and number systems in number fields by Gittenberger and Thuswaldner. The present paper is a generalization of these results to arbitrary additive functions in canonical number systems.

AMS Subject Classification (1991): 11K16, 11R47

Keyword(s): canonical number systems, $b$-additive functions, moment function

Received January 23, 2012, and in revised form April 11, 2012. (Registered under 5/2012.)

Abstract. A category ${\cal K}$ is $\alpha $-determined for some cardinal $\alpha $ if any class of non-isomorphic ${\cal K}$-objects having isomorphic endomorphism monoids is a set with fewer than $\alpha $ elements. An $\alpha $-expansion ${\cal K}_{\alpha }$ is the category whose objects are all ${\cal K}$-objects augmented by $\alpha $ new constants and whose morphisms are exactly the ${\cal K}$-morphisms preserving these constants. And a category is alg-universal if it contains an isomorphic copy of any category of algebras as a full subcategory. This paper characterizes the finitely generated varieties of distributive double $p$-algebras which are $\alpha $-determined for some cardinal $\alpha $ as well as those having $\alpha $-expansions which are alg-universal. Results of this paper complete the project of a structural classification of finitely generated varieties of distributive double $p$-algebras according to their categorical properties.

AMS Subject Classification (1991): 18B15

Keyword(s): distributive $dp$-algebra, finitely generated variety of $dp$-algebras, relatively full embedding, relative alg-universality, determinacy, Priestley duality, expansion by nullary operations

Received August 2, 2011, and in revised form January 23, 2012. (Registered under 39/2011.)

Abstract. We consider the trigonometric series $\sum_{m\in{\msbm Z}} c_m e^{imx}$, where $\{c_m: m\in{\msbm Z}\} $ is a sequence of complex numbers such that $\sum_{m\in{\msbm Z}} |c_m| < \infty.$ Then the trigonometric series converges absolutely and uniformly. We denote by $f(x)$ its sum, which is clearly continuous. We give sufficient conditions in terms of certain means of $\{c_m\} $ to ensure that $f(x)$ belongs to one of the Zygmund classes $\mathop{\rm Zyg} (\alpha )$ and zyg$(\alpha )$, where $0< \alpha\le 2$. Our theorems generalize the corresponding result of Zygmund [2] given in the special case $\alpha =1$. Our proof is essentially different from that of Zygmund. We establish two lemmas which reveal interesting interrelations between the order of magnitude of certain initial means and that of certain tail means of the sequence $\{c_m\} $.

AMS Subject Classification (1991): 26A16, 42A16

Keyword(s): trigonometric series, absolute convergence, Lipschitz classes $\mathop{\rm Lip} (\alpha )$ and lip$(\alpha )$, $0< \alpha\le 1$, Zygmund classes $\mathop{\rm Zyg} (\alpha )$ and zyg$(\alpha )$, $0< \alpha\le 2$

Received November 21, 2011, and in revised form February 24, 2012. (Registered under 65/2011.)

Abstract. Let the function $f\colon\overline {\msbm R}^2_+ \to{\msbm C}$ be such that $f\in L^1_{\rm loc} (\overline{\msbm R}^2_+)$. We investigate the convergence behavior of the double integral $(*)$\hskip25pt$\int ^{A\strut }_{0\strut }\int ^B_0 f(u,v) du dv {\rm as} A,B \to\infty $, where $A$ and $B$ tend to infinity independently of one another, while using two notions of convergence: that in Pringsheim's sense and that in the regular sense. Our main result is that if the double integral ($*$) converges in the regular sense, then the finite limits $\lim_{y\to\infty } \int ^{A\strut }_{0\strut }\left(\int ^y_0 f(u,v) dv\right ) du =: I_1 (A)$ and $\lim_{x\to\infty } \int ^B_0\left(\int ^x_0 f(u,v) du\right ) dv =: I_2 (B)$ exist uniformly in $0< A, B < \infty $, respectively, and \hskip15pt$\lim_{A\to\infty } I_1(A) = \lim_{B\to\infty } I_2 (B) =\lim_{A, B \to\infty } \int ^{A\strut }_{0\strut } \int ^B_0 f(u,v) du dv.$ This can be considered as a generalized version of Fubini's theorem on successive integration when $f\in L^1_{\rm loc} (\overline{\msbm R}^2_+)$, but $f\not\in L^1 (\overline{\msbm R}^2_+)$.

AMS Subject Classification (1991): 28A35; 40A05, 40A10, 40B05

Keyword(s): double series of complex numbers, double integrals of locally integrable functions over $\overline{\msbm R}^2_+$ in Lebesgue's sense, convergence in Pringsheim's sense, regular convergence, absolute convergence, a generalized version of Fubini's theorem on successive integration

Received April 4, 2012. (Registered under 23/2012.)

Abstract. The purpose of this paper is to deal with generalizations of ratio ergodic theorems due to R.V. Chacon, G. Baxter, and K. Jacobs. We prove two weighted generalizations of the Chacon ergodic theorem and the Jacobs random ergodic theorem. L. Sucheston has formulated a general principle yielding simultaneous proofs of many almost everywhere multiparameter convergence theorems. This principle will allow us to derive multiparameter Chacon's type ergodic theorems for positive linear contractions on $L_{1}.$ The advantage is that we can inquire further into the problem of improving the multiparameter Chacon--Ornstein ergodic theorem due to Frangos and Sucheston. A multiparameter generalization of the Dunford--Schwartz ergodic theorem is also obtained. In addition, our consideration comes to the a.e. convergence for sectorially restricted averages in the commutative case, as in the Brunel--Dunford--Schwartz theorem. Moreover, we establish two Chacon's type nonlinear ergodic theorems for the nonlinear sums of affine operators on $L_{1}$.

AMS Subject Classification (1991): 47A35, 28A35

Keyword(s): positive linear contraction, linear modulus, Sucheston principle, Chacon ergodic theorem, Jacobs random ergodic theorem, multiparameter Chacon's type nonlinear ergodic theorem, affine operator

Received July 18, 2011, and in final form February 9, 2012. (Registered under 38/2011.)

Abstract. In this paper weighted Hardy spaces where the Bohr phenomenon occurs are classified and the Bohr radius is found in terms of the generating function. Next we discuss a {\it Bohr-like phenomenon} for spaces where the Bohr phenomenon does not occur.

AMS Subject Classification (1991): 30E99

Keyword(s): Bohr phenomenon, Bohr radius, weighted Hardy spaces, generating functions

Received September 10, 2011, and in final form September 15, 2012. (Registered under 44/2011.)

Abstract. We give a Sturm-type comparison theorem and a convexity theorem for difference equations. We apply the convexity results to discrete orthogonal polynomials, such as the Hahn and Meixner polynomials by obtaining estimates on the second difference of their zeros. We show that the corresponding theorems for $q$-difference equations also hold, and present the results on the $q$-Laguerre polynomials.

AMS Subject Classification (1991): 33C45, 33D45, 39A12, 39A13

Keyword(s): Sturm comparison theorem, Sturm convexity theorem, self-adjoint equations, orthogonal polynomials, $q$-difference equations, convexity of zeros

Received January 10, 2012, and in revised form April 2, 2012. (Registered under 3/2012.)

Abstract. The boundedness of a linear or sublinear operator from the Hardy space $H_p({\msbm R}^d)$ to a quasi-Banach space is investigated.

AMS Subject Classification (1991): 42B30

Keyword(s): Hardy spaces, atomic decomposition, simple atoms, tempered distributions

Received September 27, 2011, and in revised form June 21, 2012. (Registered under 51/2011.)

Abstract. We characterize Banach lattices on which the class of order weakly compact operators coincides with that of weakly compact operators and we give some consequences.

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

Keyword(s): order weakly compact operator, weakly compact operator, order continuous norm, reflexive Banach lattice

Received July 13, 2011, and in final form April 20, 2012. (Registered under 37/2011.)

Abstract. Property $(gR)$ holds for a bounded linear operator $T\in L(X)$, defined on a complex Banach space $X$, if the isolated points of the spectrum $\sigma(T)$ of $T$ which are eigenvalues are exactly those points $\lambda $ of the approximate point spectrum such that $\lambda I-T$ is left Drazin invertible. In this paper we introduce this property and give some perturbation results.

AMS Subject Classification (1991): 47A10, 47A11; 47A53, 47A55

Keyword(s): property $(R)$, Weyl type theorems

Received February 7, 2012, and in revised form May 21, 2012. (Registered under 7/2012.)

Abstract. Given Banach spaces ${\cal X}$ and ${\cal Y}$ and operators $A\in B({\cal X})$ and $B\in B({\cal Y})$, property $(gw)$ does not in general transfer from $A$ and $B$ to the tensor product operator $A\otimes B\in B({\cal X}\overline{\otimes } {\cal Y})$ or to the elementary operator defined by $A$ and $B$, $\tau_{AB}=L_AR_B\in B(B(Y,{\cal X}))$. In this article necessary and sufficient conditions ensuring that property $(gw)$ transfers from $A$ and $B$ to $A\otimes B$ and to $\tau_{AB}$ will be given.

AMS Subject Classification (1991): 47A80, 47A53, 47A10

Keyword(s): Banach space, property $(gw)$, tensor product operator, left-right multiplication operator

Received May 28, 2011, and in revised form March 2, 2012. (Registered under 26/2011.)

Abstract. It is shown that a class of composition operators $C_\phi $ has the property that for every $\lambda $ in the interior of the spectrum of $C_\phi $ the operator $U=C_\phi -\lambda{\rm Id}$ is universal in the sense of Caradus, i.e., every Hilbert space operator has a non-zero multiple similar to the restriction of $U$ to an invariant subspace. As a generalization, weighted composition operators on the $L^2$ and Sobolev spaces of the unit interval are shown to have the same property and thus a complete knowledge of their minimal invariant subspaces would imply a solution to the invariant subspace problem for Hilbert space. Moreover, a generalization of sufficient conditions for an operator to be universal is obtained. Cyclicity and non-cyclicity results for a certain class of weights and composition functions are also proved.

AMS Subject Classification (1991): 47A15, 47B33, 47A16

Keyword(s): weighted composition operators, semi-Fredholm operators, universal operators, invariant subspaces, cyclic vectors, Müntz theorem

Received October 9, 2011, and in final form March 30, 2012. (Registered under 52/2011.)

Abstract. We develop a holomorphic functional calculus for multivalued linear operators on locally convex vector spaces, based on the resolvent identity. This includes the case of fractional powers along Lipschitz curves.

AMS Subject Classification (1991): 47A60, 47A06

Keyword(s): functional calculus, locally convex vector spaces, Lipschitz curves

Received February 11, 2009, and in revised form June 11, 2012. (Registered under 33/2009.)

Abstract. Recently, sufficent conditions for the $H^p$ boundedness of the one-dimensional Hausdorff operator were given by Liflyand and Miyachi. In this paper, we obtain new sufficent conditions for the $H^p$ boundedness of the one-dimensional Hausdorff operator. The results of Liflyand and Miyachi and the results of this paper are mutually independent. More importantly, our method in the proof allows us to study the high dimensional Hausdorff operator and fractional Hausdorff operator. We then obtain $H^p({\msbm R}^n)\rightarrow L^q({\msbm R}^n)$ and $L^p(| x| ^{\gamma }dx)\rightarrow L^q(| x| ^{\gamma }dx)$ boundedness for the high dimensional (fractional) Hausdorff operator.

AMS Subject Classification (1991): 47B38, 47D05

Keyword(s): Hausdorff operator, Hardy spaces, Marcinkiewicz interpolation, Lipschitz spaces

Received December 19, 2011, and in revised form May 2, 2012. (Registered under 67/2011.)

Abstract. The quantum variance of a self-adjoint operator depends on a density matrix whose particular example is a pure state (formulated by a projection). A general variance can be obtained from certain variances of pure states. This is very different from the probabilistic case.

AMS Subject Classification (1991): 81Q50, 40C05

Keyword(s): density matrix, pure state, projection, variance

Received November 3, 2011, and in revised form February 18, 2012. (Registered under 57/2011.)

Abstract. The problem of estimating the parameters of a linear regression model $Z(s,t)=m_1g_1(s,t)+ \cdots + m_pg_p(s,t)+U(s,t)$ based on observations of $Z$ on a spatial domain $G$ of special shape is considered, where the driving process $U$ is a Gaussian random field and $g_1, \ldots, g_p$ are known functions. Explicit forms of the maximum-likelihood estimators of the parameters are derived in the cases when $U$ is either a Wiener or a stationary or nonstationary Ornstein--Uhlenbeck sheet. Simulation results are also presented, where the driving random sheets are simulated with the help of their Karhunen--Lo?ve expansions.

AMS Subject Classification (1991): 60G60, 62M10, 62M30

Keyword(s): Wiener sheet, Ornstein--Uhlenbeck sheet, maximum likelihood estimation, Radon--Nikodym derivative

Received April 2, 2012. (Registered under 19/2012.)