Convergence of a class of means of Hp functions (0<p<1) on compact lie groups

In this paper we study the convergence of a class of means on H^p(G) (0<p<1), the means take the Bochner-Riesz means in [1], the generalized Bochner-Riesz means in [2], and the operators T^σ, in [3] as special cases. We obtain weak-type estimates for the associated maximal operators and the maximal mean boundedness for the means. Supported by NSFC     

On generalized Seiffert means

Generalizations of two Seiffert means, usually denoted by P and T, are defined and investigated. The means under discussion are symmetric and homogeneous of degree one in each variable. Computable lower and upper bounds for the new means are also established. Several inequalities involving means discussed in this paper are obtained. In particular, two Wilker’s type inequalities involving those means are derived.     

Strong monotonicity for various means

Point-wise monotonicity (in parameters) for various one-parameter families of scalar means such as power difference means, binomial means and Stolarsky means is well known, but norm comparison for corresponding operator means requires monotonicity in the sense of positive definiteness. Among other things we obtain monotonicity in the sense of infinite divisibility, which is much stronger than that in the sense of positive definiteness. These strong monotonicity results are proved based on explicit computations for measures in relevant Lévy–Khintchine (or actually Kolmogorov) formulas.     

On one-parameter family of bivariate means

A one-parameter family of bivariate means is introduced. They are defined in terms of the inverse functions of Jacobian elliptic functions cn and nc. It is shown that the new means are symmetric and homogeneous of degree one in their variables. Members of this family of means interpolate an inequality which connects two Schwab–Borchardt means. Computable lower and upper bounds for the new mean are also established.     

Sub- and superadditive integral means

The integral means are special Cauchy means (see, e.g., [L. Losonczi, On the comparison of Cauchy mean values, J. Inequal. Appl. 7 (2002) 11-24]) depending on one function. The two variable integral means were (independently) defined and studied by Elezovi? and Pe?ari? [Differential and integral f-means and applications to digamma function, Math. Inequal. Appl. 3 (2000) 189-196]. The comparison problem of two integral means (under differentiability conditions) was solved by Losonczi [Comparison and subhomogeneity of integral means, Math. Inequal. Appl. 5 (2000) 609-618]. Here we completely characterize the additive, sub- and superadditive integral means of n?2 variables.     

Invariance of means

We give a survey of results dealing with the problem of invariance of means which, for means of two variables, is expressed by the equality \(K\circ \left( M,N\right) =K\). At the very beginning the Gauss composition of means and its strict connection with the invariance problem is presented. Most of the reported research was done during the last two decades, when means theory became one of the most engaging and influential topics of the theory of functional equations. The main attention has been focused on quasi-arithmetic and weighted quasi-arithmetic means, also on some of their surroundings. Among other means of great importance Bajraktarevi? means and Cauchy means are discussed.     

Two-weight norm inequalities for the Cesàro means of Hermite expansions

An accurate estimate is obtained of the Cesàro kernel for Hermite expansions. This is used to prove two-weight norm inequalities for Cesàro means of Hermite polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted' cases. An almost everywhere convergence result is obtained as a corollary. It is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the Cesàro supremum of the means are necessary.

     

Walsh-Marcinkiewicz means and Hardy spaces

The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H _p , when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space H _p , and prove a strong convergence theorem for the Walsh-Marcinkiewicz means.     

On weighted extensions of Cauchy's means

We show that every Cauchy mean in (0,∞) can be embedded into two parameter family of weighted means. Some basic properties and examples are presented. A functional equation which appears in the problem of symmetry of these means is considered. As an application a natural extension of Stolarsky's means is obtained and a two parameter subclass of weighted power means is determined.     

N元Stolarsky平均的凸性和几何凸性

讨论了n个正数的Stolarsky平均的S-凸性和S-几何凸性,证明了:n元Stolarsky平均在r>1时是S-凸的和S-几何凸的;在r<1时是S-凹的.作为推论,此文也比较了n个正数的Stolarsky平均和算术平均的大小.     

Approximation by Means of h-Harmonic Polynomials on the Unit Sphere

The h-harmonics are analogues of the ordinary harmonics, they are orthogonal homogeneous polynomials on the sphere with respect to a weight function that is invariant under a reflection group. Two means of associated orthogonal expansions, the de la Vallée Poussin means and an analog of spherical means, are defined and their approximation behaviors are studied. A weighted modulus of smoothness is defined using the modified spherical means and is proved to be equivalent to a weighted K-modulus defined using the differential-difference h-spherical Laplacian. A Bernstein type inequality for the h-spherical Laplacian is also established.     

Riesz Means on Compact Riemannian Symmetric Spaces

We study approximation properties of the Riesz means on compact symmetric spaces of rank one. To do so we establish equivalences between the Riesz means and Peetre K-moduli and estimate the weak type and the uniform approximation of the Riesz means at the critical index. The relations between the Riesz means and the best approximation as well as the Cesàro means are also considered.     

On derivative of trigonometric polynomials and characterizations of modulus of smoothness in weighted Lebesgue space with variable exponent

Periodica Mathematica Hungarica - In this paper we investigate some properties of approximation polynomials in particular de&nbsp;la&nbsp;Vallée-Poussin means, Fejér means and...     

Run length not required: Optimal-mse dynamic batch means estimators for steady-state simulations

This paper addresses the estimation of the variance of the sample mean from steady-state simulations without requiring the knowledge of simulation run length a priori. Dynamic batch means is a new and useful approach to implementing the traditional batch means in limited memory without the knowledge of the simulation run length. However, existing dynamic batch means estimators do not allow one to control the value of batch size, which is the performance parameter of the batch means estimators. In this work, an algorithm is proposed based on two dynamic batch means estimators to dynamically estimate the optimal batch size as the simulation runs. The simulation results show that the proposed algorithm requires reasonable computation time and possesses good statistical properties such as small mean-squared-error (mse).     

On the boundedness of the maximal operators of double Walsh-logarithmic means of Marcinkiewicz type

The main aim of this paper is to investigate the (H _p , L _p )-type inequality for the maximal operators of Riesz and Nörlund logarithmic means of the quadratical partial sums of Walsh-Fourier series. Moreover, we show that the behavior of Nörlund logarithmic means is worse than the behavior of Riesz logarithmic means in our special sense.     

On the Bochner-Riesz means of critical order

Stein's well-known logarithmic asymptotics of the Lebesgue constants of the Bochner-Riesz means of critical order is extended to Lebesgue constants of more general linear means of multiple Fourier series. These means are generated by certain class of functions supported in convex domains with boundaries of non-vanishing Gaussian curvature.

     

On the jump behavior of distributions and logarithmic averages

The jump behavior and symmetric jump behavior of distributions are studied. We give several formulas for the jump of distributions in terms of logarithmic averages, this is done in terms of Cesàro-logarithmic means of decompositions of the Fourier transform and in terms of logarithmic radial and angular local asymptotic behaviors of harmonic conjugate functions. Application to Fourier series are analyzed. In particular, we give formulas for jumps of periodic distributions in terms of Cesàro–Riesz logarithmic means and Abel–Poisson logarithmic means of conjugate Fourier series.     

Knowledge trees in complex knowledge bases

We extend the use of knowledge trees as a means for questioning knowledge bases with linguistic information. Using Zadeh's theory of approximate reasoning as a tool we provide means for questioning large knowledge bases which have relational, implicational and data type information. We provide a means for answering questions of truth as well as questions of value.     

Multi-variable weighted geometric means of positive definite matrices

We define a family of weighted geometric means {G(t;ω;A)}_t∈[0,1]ⁿ where ω and A vary over all positive probability vectors in Rⁿ and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0_n) and BMP (t=1_n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.     

Estimates from Below for Lebesgue Constants

Estimates from below for the norms of linear means of multiple Fourier series are obtained. These means are given by some function λ and generalize the well-known Bochner-Riesz means. Sharpness of these estimates is established. The assumptions on λ are rather weak and of local character. Our results contain as particular cases a number of earlier published results. Proofs are based on the authors' new results on asymptotics of the Fourier transform of piecewise-smooth functions. Some applications of the results obtained are given, namely, orders of growth of the Lebesgue constants for "ovals" and "hyperbolic crosses" are evaluated and sharp conditions on the modulus of smoothness of a function are given, for this function to be approximated by the linear means.