A Cantor-Lebesgue theorem with variable ``coefficients'

If is a lacunary sequence of integers, and if for each , and are trigonometric polynomials of degree then must tend to zero for almost every whenever does. We conjecture that a similar result ought to hold even when the sequence has much slower growth. However, there is a sequence of integers and trigonometric polynomials such that tends to zero everywhere, even though the degree of does not exceed for each . The sequence of trigonometric polynomials tends to zero for almost every , although explicit formulas are developed to show that the sequence of corresponding conjugate functions does not. Among trigonometric polynomials of degree with largest Fourier coefficient equal to , the smallest one ``at' is while the smallest one ``near' is unknown.

     

二元Lagrange三角插值多项式在Orlicz空间内的逼近

研究了二元函数用一种组合型的三角插值多项式算子逼近的问题.借助连续模这一工具,给出了这类三角插值多项式在Orlicz空间内的逼近定理.     

Some alternative principles and mixed minimax theorems involving two functions

Some alternative principles and quantitative-topological minimax theorems involving two functions are given. Consequently, Geraghty–Lin's minimax theorem and Terkelsen's minimax theorem are generalized. This revised version was published online in June 2006 with corrections to the Cover Date.     

一维p-laplacian方程的非振动性

建立了一维p-laplacian方程(1)的一切解均为非振动的必要条件．所得定理改进了Kusano等在文[4]中的相应结果．     

A Condition for Convergence in Mean of Trigonometric Series

We construct an example of a trigonometric series showing that a certain condition closely related to convergence in mean of trigonometric series cannot be sharpened.     

Strong limit theorems for quasi-orthogonal random fields

This is a systematic and unified treatment of a variety of seemingly different strong limit problems. The main emphasis is laid on the study of the a.s. behavior of the rectangular means ζ_mn = 1/(λ₁(m) λ₂(n)) Σ_i=1^m Σ_k=1ⁿ X_ik as either max{m, n} → ∞ or min{m, n} → ∞. Here {X_ik: i, k ≥ 1} is an orthogonal or merely quasi-orthogonal random field, whereas {λ₁(m): m ≥ 1} and {λ₂(n): n ≥ 1} are nondecreasing sequences of positive numbers subject to certain growth conditions. The method applied provides the rate of convergence, as well. The sufficient conditions obtained are shown to be the best possible in general. Results on double subsequences and 1-parameter limit theorems are also included.     

Demystification of the geometric Fourier transforms and resulting convolution theorems

T. Qian As it will turn out in this paper, the recent hype about most of the Clifford–Fourier transforms is not thoroughly worth the pain. Almost everyone that has a real application is separable, and these transforms can be decomposed into a sum of real valued transforms with constant multivecor factors. This fact makes their interpretation, their analysis, and their implementation almost trivial. Copyright © 2015 John Wiley & Sons, Ltd.     

Razumikhin type boundedness theorems in terms of two measures for impulsive integro-differential systems

Some new direct criteria of boundedness in terms of two measures for impulsive integro-differential systems with fixed moments of impulse effects are established by Lyapunov functions coupled with Razumikhin techniques.     

均值有界变差条件及其在Fourier分析中的应用

本文宣布了关于Fourier (三角)级数的系数数列单调性条件的一个最终推广.我们已经证明了该条件对正弦级数保持一致收敛性方面已不可作任何推广,一些Fourier分析中重要和有趣的经典定理也已经在此最终条件下重新建立.本文因而对长达90年的研究历史作了一个简要综述.这一系列论文中的第一篇原始论文公布在arXiv:math.CA/0611805 v1, November 27, 2006.     

Approximation to the distribution of the least squares estimators in two dimensional cosine models by randomly weighted bootstrap

Recently, Kundu and Gupta （Metrika, 48：83 C 97, 1998） established the asymptotic normality of the least squares estimators in the two dimensional cosine model. In this paper, we give the approximation to the general least squares estimators by using random weights which is called the Bayesian bootstrap or the random weighting method by Rubin （Annals of Statistics, 9：130 C 134, 1981） and Zheng （Acta Math. Appl. Sinica （in Chinese）, 10（2）： 247 C 253, 1987）. A simulation study shows that this approximation works very well.     

Generalized integral operators and Schwartz kernel type theorems

In analogy to the classical Schwartz kernel theorem, we show that a large class of linear mappings admits integral kernels in the framework of Colombeau generalized functions. To do this, we introduce new spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that, in some sense, Schwartz' result is contained in our main theorem.     

Krasnoselskii type fixed point theorems and applications

In this paper, we establish two fixed point theorems of Krasnoselskii type for the sum of , where is a compact operator and may not be injective. Our results extend previous ones. As an application, we apply such results to obtain some existence results of periodic solutions for delay integral equations and then give three instructive examples.

     

Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings

In this paper, the famous Banach contraction principle and Caristi's fixed point theorem are generalized to the case of multi-valued mappings. Our results are extensions of the well-known Nadler's fixed point theorem [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-487], as well as of some Caristi type theorems for multi-valued operators, see [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188; J.P. Aubin, Optima and Equilibria. An Introduction to Nonlinear Analysis, Grad. Texts in Math., Springer-Verlag, Berlin, 1998, p. 17; S.S. Zhang, Q. Luo, Set-valued Caristi fixed point theorem and Ekeland's variational principle, Appl. Math. Mech. 10 (2) (1989) 111-113 (in Chinese), English translation: Appl. Math. Mech. (English Ed.) 10 (2) (1989) 119-121], etc.     

Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications

We use the hybrid method in mathematical programming to obtain strong convergence to common fixed points of a countable family of quasi-Lipschitzian mappings. As a consequence, several convergence theorems for quasi-nonexpansive mappings and asymptotically κ-strict pseudo-contractions are deduced. We also establish strong convergence of iterative sequences for finding a common element of the set of fixed point, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain the corresponding results due to Tada-Takahashi and Nakajo-Shimoji-Takahashi.     

Ferenc Lukács type theorems in terms of the Abel-Poisson mean of conjugate series

A theorem of Ferenc Lukács determines the generalized jumps of a periodic, Lebesgue integrable function in terms of the partial sum of the conjugate series to the Fourier series of . The main aim of this paper is to prove an analogous theorem in terms of the Abel-Poisson mean. We also prove an estimate of the partial derivative (with respect to the angle) of the Abel-Poisson mean of an integrable function at those points at which is smooth. Finally, we reveal the intimate relation between these two results.

     

Absolute and uniform convergence of eigenfunction expansions of integral operators with kernels admitting derivative discontinuities on the diagonals

An analog of Szasz’s theorem on the absolute convergence of trigonometric Fourier series is established for expansions in the eigen and associated functions of integral operators some of whose kernels involve derivatives with discontinuities on the diagonals.     

A Wiener-Tauberian and a Pompeiu type theorems on the Laguerre hypergroup

A Wiener-Tauberian theorem is proven on the Laguerre hypergroup [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. As consequence of this theorem we establish a Pompeiu type-theorem and we study some of its applications.     

Flag-transitive point-primitive 2-（v,k,4） symmetric designs and two dimensional classical groups

Let D be a 2-(v, k, 4) symmetric design and G be a flag-transitive point-primitive automorphism group of D with X ⊴ G ≤ Aut(X) where X ≅ PSL ₂(q). Then D is a 2-(15, 8, 4) symmetric design with X = PSL ₂(9) and X _x = PGL ₂(3) where x is a point of D.     

Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables

A new generalization of Bernstein-Stancu type polynomials for one and two variables are constructed and the theorems on convergence and the degree of convergence are established. In addition some numerical examples, corresponding to obtaining results are given.