The absolute convergence of the Fourier-Haar series for two-dimensional functions

It is well-known that if an one-dimensional function is continuously differentiable on [0, 1], then its Fourier-Haar series converges absolutely. On the other hand, if a function of two variables has continuous partial derivatives f _x ^′ and f _y ^′ on T ², then its Fourier series does not necessarily absolutely converge with respect to a multiple Haar system (see [1]). In this paper we state sufficient conditions for the absolute convergence of the Fourier-Haar series for two-dimensional continuously differentiable functions.     

On inequalities of Lieb-Thirring type

Applying the method proposed by Kashin for proving inequalities of Lieb-Thirring type for orthonormal systems, we prove a similar inequality in the multidimensional case.     

Reconstructing Coefficients of Series from Certain Orthogonal Systems of Functions

We consider a series with respect to a multiplicative Price system or a generalized Haar system and assume that the martingale subsequence of its partial sums converges almost everywhere. In this paper we prove that, under certain conditions imposed on the majorant of this sequence, the series is a Fourier series in the sense of the A-integral (or its generalizations) of the limit function if the series is considered as a series with respect to a system with supp _n < . In similar terms, we also present sufficient conditions for a series to be a Fourier series in the sense of the usual Lebesgue integral. We give an example showing that the corresponding assertions do not hold if supp _n = .     

紧拓扑半群上概率测度卷积序列的极限性质

本文讨论紧拓扑半群上概率测度卷积序列的若干重要极限性质．在第１节中，我们讨论测度集的代数结构与其支撑集代数结构的关系．第２节的定理１，通过支撑集的代数结构给出组合收敛测度序列的一个极限定理．在定理２中我们讨论独立同分布时的情形，建立了一类紧半群上的Ｋａｗａｄａ－Ｉｔ型结果．这些定理推广了紧群、紧交换半群、紧Ｌ－Ｘ半群上一些相应的结论．     

Some stochastic processes related to random density functions

If { _n } is an orthonormal system and {a _n} is a sequence of random variables such that _n (a _n )²=1 a.s. thenf(t)=| _n a _{n n} (t)|² produces a randomly selcted density function. We study the properties off under the assumptions that |a _n| is decreasing to zero at a geometric rate and { _n } is one of the following four function systems: trigonometric Jacobi, Hermite, or Laguerre. It is shown that, with probability one,f is an analytic function,f has at most a finite number of zeros in any finite interval, and the tail off goes to zero rapidly.     

The quartile operator and pointwise convergence of Walsh series

The bilinear Hilbert transform is given by

It satisfies estimates of the type

In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in , with converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.

     

Lower Bounds for n-Term Approximations of Plane Convex Sets and Related Topics

In this paper, we establish lower bounds for n-term approximations in the metric of L ²⁽I ² ) of characteristic functions of plane convex subsets of the square I ² with respect to arbitrary orthogonal systems. It is shown that, as n, these bounds cannot decrease more rapidly than .     

A complete orthonormal system of divergence

A complete orthonormal system of functions {Θ_n}_n=1^∞,Θ_n∈L_[0,1]^∞, defined on the closed interval [0,1] is constructed such that ∑_n=1^∞a_nΘ_n diverges almost everywhere for any .For the constructed system the following result is true:Corollary 1.Any nontrivial series in the system {Θ_n}_n=1^∞which converges in measure to zero diverges almost everywhere.     

Riesz Bases Generated by Contractions and Translations of a Function on an Interval

Conditions for a system of contractions and translations of a function to be a Riesz basis are given.     

On an Inequality of A. Grothendieck Concerning Operators on 1

In 1955, A. Grothendieck proved a basic inequality which shows that any bounded linear operator between L¹(µ)-spaces maps (Lebesgue-) dominated sequences to dominated sequences. An elementary proof of this inequality is obtained via a new decomposition principle for the lattice of measurable functions.     

On some integral properties of multidimensional Hilbert operators

In the present paper, we study the integral properties of multidimensional Hilbert transforms.     

Lacunary convergence of series in

For a finite measure , let denote the space of -measurable functions equipped with the topology of convergence in measure. We prove that a series in is subseries (or unconditionally) convergent provided each of its lacunary subseries converges.

     

Value distribution of the Lerch zeta-function with algebraic irrational parameter. III

We obtain a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions for the Lerch zeta-function with algebraic irrational parameter. Partially supported by the Lithuanian Studies and Science Foundation.     

Estimation of multivariate binary density using orthogonal functions

In this paper, the authors studied certain properties of the estimate of Liang and Krishnaiah (1985, J. Multivariate Anal. 16, 162–172) for multivariate binary density. An alternative shrinkage estimate is also obtained. The above results are generalized to general orthonormal systems.     

Concentration of Haar measures, with an application to random matrices

We show that the mixing times of random walks on compact groups can be used to obtain concentration inequalities for the respective Haar measures. As an application, we derive a concentration inequality for the empirical distribution of eigenvalues of sums of random Hermitian matrices, with possible applications in free probability. The advantage over existing techniques is that the new method can deal with functions that are non-Lipschitz or even discontinuous with respect to the usual metrics.     

A combination of penalty function and multiplier methods for solving optimal control problems

The properties of combined multiplier and penalty function methods are investigated using a second-order expansion and results known for the Riccati equation. It is shown that the lower bound of the values of the penalty constant necessary to obtain a minimum is given by a certain Riccati equation. The convergence rate of a common updating rule for the multipliers is shown to be linear.This work has been supported by the Swedish Institute of Applied Mathematics.     

Convergence rate of a proximal multiplier algorithm for separable convex minimization

In this paper, we analyse the convergence rate of the proximal algorithm proposed by us in the article [A proximal multiplier method for separable convex minimization. Optimization. 2016; 65:501–537], which has been proposed to solve a separable convex minimization problem. We prove that, under mild assumptions, the primal-dual sequences of the algorithm converge linearly to the optimal solution for a class of proximal distances.     

Value distribution of general Dirichlet series. VII

We prove a limit theorem in the space of meromorphic functions for a new class of general Dirichlet series. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 193–202, April–June, 2006.     

一种新的迭代收敛阶数的证明与推广

迭代方法是求解非线性方程近似根的重要方法.本文基于隐函数存在定理,提出了一种新的迭代方法收敛性和收敛阶数的证明方法,并分别对牛顿(Newton)和柯西(Cauchy)迭代方法迭代收敛性和收敛阶数进行了证明.最后,利用本文提出的证明方法,证明了基于三次泰勒(Taylor)展式构成的迭代格式是收敛的,收敛阶数至少为4,并提出猜想,基于n次泰勒展式构成的迭代格式是收敛的,收敛阶数至少为(n+1).