ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. The purpose of this note is to study the relationship between the Besov spaces $B_p$ (or $B_p^\#$) and the recently introduced classes of functions $Q_{q,0}$ (or $Q_{q,0}^\#$). We establish necessary and sufficient conditions for a lacunary series to belong to $B_p$ and use this result to show that our generalizations of two basic theorems involving Besov spaces are best possible. There are some unexpected differences between the results for analytic Besov spaces, $B_p$, and those for spherical Besov spaces, $B_p^\#$; these differences arise when $1< p< 2$. As a by-product of these results we also obtain a new characterization of the family of little normal functions.

AMS Subject Classification (1991): 30H05, 30B10, 30D45

Received March 16, 1994 and in revised form October 19, 1994. (Registered under 5617/2009.)

Abstract. Complementary spaces for Fourier series were introduced by G. Goes and generalized by M. Tynnov. In this paper we give a notion of complementary spaces for Fourier transforms treated as distributions. Applications to multipliers and summability are given. Some known conditions turn out to be effective for $(L^p,L^p)$ multipliers with $1 \le p \le\infty $.

AMS Subject Classification (1991): 42A38, 42A24, 42A45

Keyword(s): Fourier transform, complementary space, summability, multiplier

Received September 9, 1994 and in revised form January 13, 1995. (Registered under 5618/2009.)

Abstract. Kramer's sampling theorem, a generalization of Shannon's sampling theorem, states that a function $f$ which is representable as a finite integral transform can be reconstructed from sample values $f(t_k)$ in terms of a series expansion with respect to a complete orthogonal set. The aim of this paper is to investigate the error occurring when this expansion is used for a function $f$ which is representable as an infinite rather than as a finite integral transform. In particular, it is shown that in many applications this error tends to zero when the distance between the sampling points $t_k$ tends to zero.

AMS Subject Classification (1991): 94A12, 41A25, 41A58, 34B24

Received October 27, 1994, and in revised form February 28, 1995. (Registered under 5619/2009.)

Abstract. A variant of multiparameter Calderón--Zygmund theory combined with interpolation arguments is used to prove new results on homogeneous Fourier multipliers.

AMS Subject Classification (1991): 42B20, 42B15, 46B70

Keyword(s): Multiparameter Calderón--Zygmund theory, Analytic families of operators, Homogeneous multipliers

Received March 8, 1994. (Registered under 5620/2009.)

Abstract. It is shown that the realizations of the gaussian field $\{X(a), a\in{\msbm R}^d\} $ with $E(X(a)-X(b))^2=|a-b|^\beta $, $0<\beta < 2$, and mean zero, are locally in the Hölder class of exponent $\beta /2$ in the Orlicz norm corresponding to the Young function with tail $\exp(u^2)$. Other support function spaces are treated as well. In the proofs a crucial role is played by the theorem on the equivalence of modified Franklin and diamond bases in some generalized Hölder classes on a cube. Moreover, it is shown that the box dimension of the graph of $X(\cdot )$ over a cube is almost surely equal to $d+1-\beta /2$.

AMS Subject Classification (1991): 28C20, 41A15, 60B01, 60G15, 60G17

Received November 23, 1994; in revised form March 30, 1995. (Registered under 5621/2009.)

Abstract. Let the finite abelian group $G$ be the direct product of the cyclic subsets $A_1,\ldots,A_n$ with $1\in A_1\cap\ldots \cap A_n$. We prove that every quasi-perodic partial product of this product contains a subgroup among its factors. This extends a result of G. Hajós stating the existence of a subgroup among the factors of the whole product.

AMS Subject Classification (1991): 20K01, 52C22

Keyword(s): factorization of finite abelian groups, Hajós--Rédei theory

Received September 30, 1994. (Registered under 5622/2009.)

Abstract. The authors estimate the best algebraic approximation error of the locally absolutely continuous functions in the space $L_u^p([-1,1])$, where $u$ is a generalized Ditzian--Totik weight. A simultaneous approximation theorem is shown for the Fourier projector corresponding to the generalized Jacobi polynomials. Finally, the boundedness of the Lagrange interpolation operator is proved in a space of Sobolev-type.

AMS Subject Classification (1991): 41A05

Received September 28, 1994 and in revised form December 8, 1994. (Registered under 5623/2009.)

Abstract. We study continuity and moduli of continuity sample path properties of $\ell ^p$-valued Gaussian processes in general, and those of $\ell ^p$-valued Ornstein--Uhlenbeck processes in particular, for $1\leq p \leq2$.

AMS Subject Classification (1991): 60G15, 60G10, 60G17, 60F15; 60G07, 60F10

Keyword(s): Banach space valued processes, \ell ^p, -valued Gaussian processes, \ell ^p, -valued Ornstein--Uhlenbeck processes, moduli of continuity, quasi-increasing functions

Received June 15, 1994. (Registered under 5624/2009.)

Abstract. Extending the respective univariate result of Breuer and Major and of Giraitis and Surgailis, a vector process of shifted partial sums of a stationary short-range dependent sequence under instantaneous Gaussian subordination is shown to converge in distribution to the corresponding vector of independent Wiener processes. This result is then used to derive the asymptotic finite-dimensional distributions of kernel estimators of a regression function in a fixed-design nonparametric regression model with short-range dependent errors under Gaussian subordination. The asymptotic distributions of one- and two-sided maximal deviations of the kernel estimators over an increasingly finer grid are also determined when the short-range dependent errors in the model are normal.

AMS Subject Classification (1991): 62G07, 60F17

Received September 30, 1994, and in revised form March 17, 1995. (Registered under 5625/2009.)

Abstract. Given a finite partially ordered set $P$, for subsets or, in other words, coalitions $X$, $Y$ of $P$ let $X\le Y$ mean that there exists an injection $\varphi\colon X \to Y$ such that $x\le\varphi (x)$ for all $x\in X$. The set ${\cal L}(P)$ of all subsets of $P$ equipped with this relation is a partially ordered set. All partially ordered sets $P$ such that ${\cal L}(P)$ is a lattice are determined, and this result is extended to quasiordered set $P$ versus $q$-lattice ${\cal L}(P)$ as well. Some elementary properties of distributive lattices ${\cal L}(P)$ are also given.

AMS Subject Classification (1991): 06B99, 06A99, 90D99

Keyword(s): Lattice, q, -lattice, quasiorder, partially ordered set, coalition

Received July 29, 1994. (Registered under 5626/2009.)

Abstract. Given a lattice $L$, the lattice, in fact the involution lattice, $\mathop{\rm Quord}(L)$ of quasiorders of $L$ is shown to be isomorphic with $\mathop{\rm Con}^2(L)$, the direct square of the congruence lattice of $L$. The isomorphism given is natural in category theoretic sense. As a corollary, a description of compatible partial orderings of a lattice is obtained.

AMS Subject Classification (1991): 06B99, 08A30

Keyword(s): quasiorder, compatible order, lattice, involution lattice, natural equivalence

Received August 22, 1994. (Registered under 5627/2009.)

Abstract. A new type of dependencies in a relational database model is investigated. If $b$ is an attribute, $A$ is a set of attributes then it is said that $b(p,q)$-depends on $A$, in notation $A\mathop{\longrightarrow}^{(p,q)}b$, in a database $r$ if there are no $q+1$ relations in $r$ such that they have at most $p$ different values in $A$, but $q+1$ different values in $b$. $(1,1)$-dependency is the classical functional dependency. Let ${\bf J}(A)$ denote the set $\{b:A\mathop{\longrightarrow}^{(p,q)}b\}$. Using some characterization of the set function ${\bf J}(A)$ we give estimates for the minimum number of records in a database system that results the set functions ${\bf J}(A)$.

AMS Subject Classification (1991): 68P15, 68R05, 05D05

Received October 19, 1994, and in revised form April 5, 1995. (Registered under 5628/2009.)

Abstract. For the simplex $S$ and a Jacobi-type weight the Bernstein-Durrmeyer operators will be given and their properties investigated. A direct estimate and a matching strong converse inequality will be derived.

AMS Subject Classification (1991): 41A36, 41A25, 41A63

Keyword(s): Multivariate Jacobi polynomials, Bernstein--Durrmeyer operators, direct and converse inequalities

Received April 7, 1994. (Registered under 5629/2009.)

Abstract. Simultaneous rational approximants to a vector of functions $(f_1,\ldots,f_m)$ that forms a Nikishin system are investigated. A new error formula for the $m$-th component of the approximation error is proved. It is well known that the common denominators of the simultaneous approximants satisfy a multiple orthogonality relation; it is shown that they also satisfy an ordinary one, which allows us to prove interesting properties of the approximants. The zeros of the $m^{th}$ remainder function $R_m$ are analysed. Complementary results for the zeros of the remainder functions $R_1,\ldots, R_{m-1}$ will be published in a separate paper.

AMS Subject Classification (1991): 41A21, 30E10

Keyword(s): Nikishin systems, simultaneous rational approximants, normality, orthogonal polynomials, multiple orthogonality

Received June 13, 1994. (Registered under 5630/2009.)

Abstract. A characterization of the spectral maximal subspaces of translation operators in $L^p(G)$ spaces, where $G$ is locally compact abelian group and $1\le p\le\infty $, is given in terms of Fourier transforms. We apply this to the case of the group of integers for the values $1\le p< \infty $ and to the case of the space $c_0({\msbm Z})$. We show how the Riemann--Lebesgue localization principle for Fourier series and our results together imply the localization principle for conjugate series.

AMS Subject Classification (1991): 47B40, 47A11, 42A20

Received May 26, 1994. (Registered under 5631/2009.)

Abstract. The aim of this paper is to prove a duality relation between strong summability and Sidon type inequalities. We give examples which show how some fundamental results in strong summability correspond, through this duality, to Sidon type inequalities, and conversely.

AMS Subject Classification (1991): 40F05, 42A05, 41A25, 42A20

Keyword(s): strong summability, strong approximation, Sidon type inequalities, dyadic Hardy space, Orlicz spaces

Received October 12, 1994. (Registered under 5632/2009.)

Abstract. Sufficient ultraspherical multiplier criteria are refined in such a way that they are comparable with necessary multiplier conditions. Also new necessary conditions for Jacobi multipliers are deduced which, in particular, imply known Cohen type inequalities. Muckenhoupt's transplantation theorem is used in an essential way.

AMS Subject Classification (1991): 33C45, 42A45, 42C10

Keyword(s): Ultraspherical polynomials, multipliers, necessary conditions, sufficient conditions, Cohen type inequalities, fractional differences

Received September 26, 1994. (Registered under 5633/2009.)

Abstract. It is known that the Fejér means -- with respect to Vilenkin systems -- of an integrable function on bounded Vilenkin groups converge to the function a.e. In this work we prove the above on similar, not necessarily Abelian, totally disconnected groups with respect to the product system of normed coordinate functions of continuous irreducible unitary representations of the coordinate groups.

AMS Subject Classification (1991): 43A75, 43A55, 22D10

Keyword(s): compact totally disconnected groups, Fejér means, maximal function, pointwise convergence, (H, typeness, L)

Received September 26, 1994, and in revised form April 12, 1995. (Registered under 5634/2009.)

Abstract. In [1] the notion of the general product of nondeterministic automata was introduced, and the isomorphically and homomorphically complete systems were studied. It was proved that there are finite complete systems for both representations and also a characterization was given for the isomorphic case. In this work we deal with the isomorphic representation of nondeterministic automata with respect to the cube-product introduced for deterministic automata in [2]. We characterize isomorphically complete systems for this product, from which it will follow that the general product and the cube-product are equivalent with respect to the isomorphic completeness.

AMS Subject Classification (1991): 68Q

Received May 4, 1994 and in revised form September 14, 1994. (Registered under 5635/2009.)

Abstract. First, we study cosine and sine Fourier transforms. We prove that if $f$ is an absolutely continuous function over $[0,\infty )$ such that $\lim_{t\to\infty }f(t)=0$ and for some $p>1$ we have $$\int_0^\infty\left ({1\over u}\int_u^{2u}|f'(t)|^pdt\right )^{1/p}du < \infty,$$ then $(*) \hat{f}_c(x):=\int_0^{\to\infty }f(t)\cos xt dt$ converges for any $x>0, \hat{f}_c$ is Lebesgue integrable over $[0,\infty )$, and the inversion formula holds. Under the above conditions, $(**) \hat{f}_s(x):=\int_0^{\to\infty }f(t)\sin xt dt$ converges for any $x>0$, $\hat{f}_s$ is $L$-integrable over $[0,\infty )$ if and only if $f(t)/t$ is $L$-integrable over $[0,\infty )$, in which case the inversion formula holds. Our basic tools are two Sidon type inequalities which we elaborate also in this paper. Second, we extend these results to complex Fourier transforms over $(-\infty,\infty )$. Third, we deduce sufficient conditions for the $L$-integrability of trigonometric series over $[-\pi,\pi ].$ In this way, we obtain new proofs of the theorems by G.A. Fomin, J. Fournier and W. Self, which are known presently as the most general ones. Fourth, as a by-product, we obtain sufficient conditions under which a function is the complex Fourier transform of an $L$-integrable function over $(-\infty,\infty ).$

AMS Subject Classification (1991): 42A38, 26A46

Keyword(s): cosine and sine Fourier transforms, complex Fourier transform, absolutely continuous function, cosine and sine series, complex trigonometric series, Lebesgue integrability, Fourier series, Hausdorff-Young inequality, inequality of Titchmarsh, Sidon type inequalities, inversion formula

Received September 1, 1994. (Registered under 5636/2009.)

Abstract. We obtain the necessary and sufficient condition for the weak convergence of $U$-statistic type processes in weighted metrics. The results are applied to the detection of changes in two data sets.

AMS Subject Classification (1991): 62G10, 60F17

Received June 20, 1994, and in revised form February 14, 1995. (Registered under 5637/2009.)

Abstract. M. Frentiu defined certain linear combinations of Szász--Mirakjan operators. In the case of $2r$-times $(r>1)$ differentiable functions, these linear combinations provide a better approximation than the original operator. In this paper we show that this method can be applied to a certain extension of Szász--Mirakjan operators that has been introduced in [3], in order to obtain an approximation on the whole real axis.

AMS Subject Classification (1991): 41A35

Received October 2, 1994. (Registered under 5638/2009.)

Abstract. The nonlinear scalar equation $$x'(t)=b(t)f(x(t-T))-c(t)g(x(t)) (c(t)\ge0)$$ is considered under the assumption $|f(x)|\le\kappa |g(x)|$ $(|x|\le\epsilon _0)$ with appropriate constants $\kappa,\epsilon_0>0$. Sufficient conditions are given for the asymptotic stability of the zero solution by Lyapunov's direct method with Lyapunov functionals. The effect of the dominating conditions $$c(t)-\kappa |b(t+T)|\ge\mu \ge0, c(t)- \kappa |b(t)|\ge\nu \ge0$$ for all $t\ge0$ with constant $\mu $, $\nu $ is discussed by examples.

AMS Subject Classification (1991): 34K20

Received November 9, 1994. (Registered under 5639/2009.)

Abstract. There are several equivalence assertions known for the approximation of functions, continuous on an interval, by sequences of positive linear operators (e.g., operators of exponential type). Due to the improved rate of convergence near the endpoints of the interval under consideration, all these results indeed possess a pointwise (weighted) structure. In particular in connection with characterizations via the Ditzian--Totik modul, the present paper contributes to the question whether the individual direct and inverse estimates are sharp on large point sets. In fact, with the aid of an appropriate quantitative resonance principle and Borel--Cantelli-type result, the existence of counterexamples is established, simultaneously delivering the sharpness of the Jackson as well as of the Berenstein estimate on sets of full measure. To apply the general arguments mentioned, it is essential to determine relevant resonance elements. To this end, a unifying approach is suggested, based on the construction of a suitable matrix of knots.

AMS Subject Classification (1991): 41A25, 41A36

Received September 30, 1994. (Registered under 5640/2009.)

Abstract. In this paper we investigate the topological properties of the set of ``fractions'' in a ``number system'' with digits $0$ and $1$ and a complex base. It is also investigated, whether the translates of this set with ``integers'' of the same number system gives a ``just touching'' covering of the plane. Algorithms are constructed to decide such questions and machine computations are also done.

AMS Subject Classification (1991): 28A80, 54-04, 11K16, 68Q20

Received June 13, 1994. (Registered under 5641/2009.)