ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. The aim of this paper is to generalize the results of [2]. We determine and prove unicity of polynomial $R(x)$ of the lowest possible degree, which has the interpolation properties $R(x_k)=y_k$ ($k=1,\ldots,n$) and $R'(x^*_k)=y_k^\prime $ ($k=1,\ldots,n$), where $x_k$'s are any distinct real nodal points generating the polynomial $w(x)$ and $x_k^*$'s are the roots of $aw(x)+bw'(x)$, where $a$, $b$ are any real numbers, $a\not =0$, $b\not =0$. The case $b=0$ is the Hermite--Fejér interpolation and the case $a=0$ is the Pál interpolation.

AMS Subject Classification (1991): 41A05

Received December 23, 1993; in revised form March 24, 1994. (Registered under 5642/2009.)

Abstract. Extending results by B.P. Duggal in [4], structure of essentially normal roots of normal operators is investigated.

AMS Subject Classification (1991): 47B15, 47A55

Received October 7, 1994. (Registered under 5643/2009.)

Abstract. The present paper follows a previous work where energy decay estimates were obtained for strong solutions of a Petrovsky system with nonlinear internal damping. The well-posedness of the problem and the estimates were obtained by using the Faedo--Galerkin method. In this paper we apply a semigroup approach. This also allows us to consider weak solutions and to remove some unnatural technical assumptions concerning the nonlinearities. Moreover, this method leads to shorter proofs.

AMS Subject Classification (1991): 93C20, 93D15, 35Q72

Received November 7, 1994. (Registered under 5644/2009.)

Abstract. Continuing the investigation of Müntz type problems for Bernstein polynomials $x^{k_j}(1-x)^{n-k_j}$ started in [3], we introduce the notion of density of the sequence $\{k_j\} $, and prove convergence theorems when this density is at least $1/2$. A particular borderline-case ($k_j=2j$) is reformulated and solved in terms of self-reciprocal polynomials, whose approximation theoretic properties are of independent interest. Approximation by positive linear combinations of the above mentioned Bernstein polynomials is also considered.

AMS Subject Classification (1991): 41A10, 41A17, 41A29

Received April 29, 1994. (Registered under 5645/2009.)

Abstract. It is proved that functions are determined by their integrals over rotation invariant families of hypersurfaces.

AMS Subject Classification (1991): 44A12, 45D055

Keyword(s): Radon transform, spherical harmonics

Received November 03, 1993. (Registered under 5646/2009.)

Abstract. We examine the problem of equivalence of the following conditions: $$\sum^\infty_{m=0} \alpha_m\Bigl(\sum^{\nu_{m+1}}_{n=\nu_m+1}|c_n|^q\Bigr)^{p/q}< \infty\hbox{ and } \sum^\infty_{m=1} \beta_m \Bigl(\sum^m_{n=1} \gamma_n|c_n|^q\Bigr)^{p/q} < \infty.$$ Using these results and some known theorems we give new conditions for $|R,\lambda,\gamma(t)|_k$-summability of orthogonal series, furthermore some structural conditions pertaining to Fourier series.

AMS Subject Classification (1991): 40A05, 42A28, 42C15

Received February 22, 1994. (Registered under 5647/2009.)

Abstract. The approximation in the strong sense of continuous functions by the subsequences of the triangular partial sums of their Fourier series are investigated. The strong means are constructed by a sum function instead of the power function.

AMS Subject Classification (1991): 42B08

Received September 13, 1994. (Registered under 5648/2009.)

Abstract. We investigate converse quadrature sum (or Marcinkiewicz--Zygmund) inequalities of the form $$\int_{-\infty}^\infty |PW|^p(x)\bigl(1+|x|\bigr)^{rp}dx\le C\sum_{j=1}^n \lambda_{jn}|PW|(x_{jn})^p W^{-2}(x_{jn})\bigl(1+|x_{jn}|\bigr)^{Rp} $$ for all polynomials $P$ of degree $\le n-1$ and $n\ge1$, with $C$ independent of $n$ and $P$. Here the $\{x_{jn}\}$ and $\{\lambda_{jn}\}$ are the Gauss points and weights for the Freud weight $W^2$ (for example, $W_\beta(x) :=\exp\left(-\frac12 |x|^\beta\right)$, $\beta >1$) and $r,R\in{\msbm R}$, $1< p< \infty$. We derive necessary and almost matching sufficient conditions for such inequalities that may be applied at least for $W^2=W_\beta^2$, $\beta >1$.

AMS Subject Classification (1991): 41A55, 65D04, 42C05

Keyword(s): Gauss quadrature, quadrature sums, polynomials, Freud weights, Lagrange interpolation, orthogonal expansions

Received July 27, 1994. (Registered under 5649/2009.)

Abstract. The $\alpha $-duals of the sequence spaces $w_{\infty } (p)$ and $w_{0}(p)$ were determined in [2], Satz 8, p. 79, and [1], Th. 4, p. 103 for $p_{k}\le1$ ($k=1,2,\ldots$). Nothing seems to be known in the case $p_{k}>1$ ($k=1,2,...$) except when $p_{k}=p>1$ for all $k$ (cf. [3]). Here we shall determine the $\alpha $-duals of $w_{\infty }(p)$ and $w_{0}(p)$ in some cases where $p_{k}>1$ ($k=1,2,...$).

AMS Subject Classification (1991): 40H05

Keyword(s): sequence spaces, dual spaces

Received September 19, 1994, in revised form February 8, 1995. (Registered under 5650/2009.)

Abstract. Recently L. Leindler proved the converse of some inequalities of Hardy--Littlewood type. Now we give a certain generalization of one of these theorems giving with this the converse of some inequalities proved by the present author earlier.

AMS Subject Classification (1991): 40A05

Received October 21, 1994. (Registered under 5651/2009.)

Abstract. The mentioned inequalities in the title are investigated in metric $C(\Gamma )$, where $\Gamma $ is a manifold described by trigonometric polynomials.

AMS Subject Classification (1991): 41A10

Keyword(s): Approximation by polynomials

Received September 28, 1994. (Registered under 5652/2009.)

Abstract. Let ${{\msbm T}}={{\msbm R}}/2\pi{{\msbm Z}}$ be the circle group, ${\bf C}({{\msbm T}})$ be the set of continuous functions on ${\msbm T}$, and ${\cal T}$ be the set of the trigonometrical polynomials. Let $f\in{\cal T}$ be nonnegative, even and positive (semi)definite. In number theory and analysis itself various extremal problems are related to the determination of the least or largest possible value of $f(0)$ under various conditions on the degree, spectrum set, or the value of some prescribed coefficients. We define $$\eqalign{{\cal F}(a)=\{f\in{\bf C}({{\msbm T}}):f(x)\sim1+a\cos x+\sum_{k=2}^\infty a_k &\cos kx\geq0 (\forall x),\cr &a_k\geq0 (k\in{{\msbm N}})\}}$$ and the extremal quantity $$\alpha(a)=\inf\lbrace f(0): f\in{\cal F}(a)\rbrace.$$ The aim of the paper is to collect as much information about $\alpha(a)$ as possible. Although the motivation of studying $\alpha(a)$ stems from the application of $\alpha(a)$ to extremal problems, e.g. one posed by Landau, we do not describe these connections here. Let us only mention that the new results of the paper, in particular concerning the behavior of $\alpha$ at $a\to2$, provide essential help in those questions. However, the general approach here, and in particular the precise answer to the problem of estimation of Fourier coefficients of a nonnegative Fourier series from information regarding the first coefficient, as formulated in Theorem 2.1, seems to be of independent interest. Special emphasis is given to the description of $\alpha(a)$ at $a\to2$. This analysis leads to considerable improvements upon earlier results of French and Steckin. However, our results settle only the order of magnitude of $\alpha(a)$, and a precise asymptotic description remained an open question.

AMS Subject Classification (1991): 42A05, 42A82, 41A15, 46A55

Received October 27, 1994. (Registered under 5653/2009.)

Abstract. For general orthogonal series and nonnegative summability methods we consider the interrelation between the rate of convergence of strong means and of certain partial sums. Therefore Toeplitz-like conditions on the method are introduced and their strength is discussed.

AMS Subject Classification (1991): 42C15

Received January 4, 1995. (Registered under 5654/2009.)

Abstract. Partial Jordan-triples are algebras with three variables occuring naturally in the description of the holomorphic automorphism groups of bounded circular domains. The following question is studied: under which conditions can all inner derivations of a partial Jordan-triple be recovered from their restrictions to the complete algebraic base of the triple.

AMS Subject Classification (1991): 17C65, 17C30, 32M05, 32M15, 32H20

Received October 19, 1994. (Registered under 5655/2009.)

Abstract. Let $A^p$ and $S^p$, $p\ge1$, denote the spaces of functions whose Fourier series are absolutely, respectively strongly convergent a.e. of index $p\ge1$. They are Banach spaces; $A^p\subset S^p\subset\cap _{r\ge1}L^p$ for $p>1$, and $A^1={\cal A}$ is the classical space of absolutely convergent trigonometric series. It is well known that $\tilde{\cal A}=\ell ^1$ so that the dual space $({\cal A})^*$ can be identified with the space of Radon measures. We extend this statement to the function spaces $A^p$, $p>1$ by showing that $(A^p)^*$, the dual of $A^p$, can be identified with $A^q$, $1/p+1/q=1$. Hence, the spaces $A^p$, $p\ge1$ are reflexive. We prove this by observing that $A^p$, $p\ge1$ can be characterized in terms of the sequences of Fourier coefficients and applying the sequence space approach. Similarly, giving various description of the spaces $\widehat{S^p}$, we examine the duals of $S^p$ for $p>1$ and show that these spaces are not reflexive. The dual space $(S^p)^*$ can be identified with the space $\{g\in L^1:\sum_{j=0}^\infty\|d^jg\|_{S^q}< \infty\} $, where $d^jg$ denotes the $j$-th dyadic section of $g$. All of these spaces are examples of the so called mixed norm spaces.

AMS Subject Classification (1991): 42A16, 42A20

Received September 28, 1994. (Registered under 5656/2009.)

Abstract. Let $\bf A$ be a strictly simple algebra generating a locally finite minimal variety, and let us expand $\bf A$ arbitrarily with new operations to get an algebra ${\bf A}^\bullet $. We investigate the question under what conditions ${\bf A}^\bullet $ generates a minimal variety. Our result shows that if the tame congruence theoretic type label of $\bf A$ is distinct from 5 or if $\bf A$ has a trivial automorphism group, then ${\bf A}^\bullet $ generates a minimal variety if and only if ${\bf A}^\bullet $ is nonabelian or has a trivial subalgebra.

AMS Subject Classification (1991): 08B05, 08A05, 08A40

Received October 25, 1994, and in revised form January 17, 1995. (Registered under 5657/2009.)

Abstract. Gallai's min--max theorem for interval systems has been derived from Brouwer's fixed point theorem.

AMS Subject Classification (1991): 05C70, 55M20

Received August 26, 1994. (Registered under 5658/2009.)

Abstract. In this paper we examine approximation on $B_{pq}^{\alpha }$, Besov spaces over the $2$-series field, for ${1\over2} < p< 1$. Namely, we shall obtain strong and weak estimates for certain operators associated with strong approximation by the Cesàro means, the strong Cesàro means, and a transplantation theorem for maximal multipliers on $B_{pq}^{\alpha }$. We also estimate the operator norms of partial sums and Cesàro means, and use this estimate to characterize Besov Walsh--Fourier series.

AMS Subject Classification (1991): 42C10, 26A16, 42A24, 42A45

Keyword(s): Walsh functions, Besov spaces, strong approximation, strong Cesàro means, transplantation

Received August 22, 1994. (Registered under 5659/2009.)

Abstract. The pantograph equation $$\dot x(t)=A(t)x(t)+B(t)x(\theta(t))$$ is considered, where $A$, $B$ are continuous matrix functions, the lag function $\theta\colon {\msbm R}_+\to{\msbm R}_+$ is continuous and $0\le\theta (t)\le t $ for all $t\ge0$. A representation using the Cauchy matrix of the system $\dot x=A(t)x$ is given for the solutions. This representation gives the well-known Dirichlet series solution for the scalar ``autonomous" cases $$\dot x(t)=ax(t)+bx(pt)\qquad (a, b, p \hbox{ are constants, }0< p< 1).$$

AMS Subject Classification (1991): 34K05

Received January 18, 1995. (Registered under 5660/2009.)

Abstract. It is proven, that for Cantor type sets the Markoff property (i.e. the fact that the Markoff constants are of polynomial growth) is equivalent to the Hölder continuity property of the associated Green function, which in turn is equivalent to the fact that the length of the generating intervals at level $n$ is not smaller than $\alpha ^n$ for some $\alpha >0$. It is also shown that for regular Cantor sets the fastest possible increase of the Markoff constants is subexponential.

AMS Subject Classification (1991): 26B10, 31A15

Received October 19, 1994, and in revised form March 21, 1995. (Registered under 5661/2009.)

Abstract. We consider Hermite-Fejér type interpolation based on $\rho $-normal system and give the convergence order for continuous function.

AMS Subject Classification (1991): 41A05, 41A10

Received September 26, 1994. (Registered under 5663/2009.)

Abstract. Let $X_{1,n}\le\cdots \le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ belonging to the domain of attraction of an extreme value distribution and let $k_n$ be positive integers such that $k_n\to\infty $ and $k_n/n\to\alpha $ as $n\to\infty $, where $0\le\alpha < 1$. Given known constants $d_{i,n}$, $1\le i\le n$, that are all specified by the statistician, consider the linear combination $T_n(k,k_n)=\sum_{i=k+1}^{k_n} d_{n+1-i,n}f(X_{n+1-i,n})$ of extreme values, where $k\ge0$ is any fixed integer and $f$ is a Borel-measurable function. With suitable $f$ and normalizing and centering constants $A_n>0$ and $C_n$ we determine the limiting distribution of the sequence $(T_n(k,k_n)-C_n)/A_n.$ Using linear combinations of extreme values, we also generalize the classical Hill estimator for the index of a distribution function $F$ with a regularly varying upper tail.

AMS Subject Classification (1991): 62E20; 62G05, 62G30

Received June 21, 1994, and in revised form October 15, 1994. (Registered under 5664/2009.)

Abstract. The Burkholder--Gundy inequality is shown for vector valued martingales. Using this we extend a classical result due to Marcinkievicz and Zygmund to the two-parameter Walsh-system. We extend some results of Sunouchi from one dimension to two dimensions and, with the help of martingale theory, we prove that the Sunouchi operators and the supremum operator of the strong $(C,\alpha,\beta,q)$ means are bounded operators from $L_p$ to $L_p$ $(1< p< \infty )$. As a consequence it is obtained that every function $f\in L_p$ $(1< p< \infty )$ is strong $(C,\alpha,\beta,q)$ summable.

AMS Subject Classification (1991): 42C10, 43A75, 40F05, 60G42

Received September 21, 1994. (Registered under 5665/2009.)