GENERALIZED SPHERICAL RIESZ MEANS OF MULTIPLE FOURIER SERIES

Let Q be the k-dimensional fundamental cube and L(Q)denote the space ofall functions f integrable on Q and periodic in each variable.The Fourier seriesof f∈L(Q)is σ(f)(x)～∑a_m(f)e~(imx). (1)     

APPROXIMATION OF SPHERICAL MEANS OF MULTIPLE FOURIER INTEGRALS AND FOURIER SERIES ON SET OF FULL MEASURE

In this paper we consider the approximation for functions in some subspaces of L~2 by spherical meansof their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.     

UNIFORM CONVERGENCE OF CESARO MEANS OF NEGATIVE ORDER OF DOUBLE TRIGONOMETRIC FOURIER SERIES

In this paper we prove that iff ∈ C([-π,π]2) and the function f is bounded partial p-variation for some p ∈ [1, ∞), then the double trigonometric Fourier series of a function f is uniformly (C;-α,-β) summable (α β＜ 1/p,α,β＞ 0) in the sense of Pringsheim. If α β≥ 1/p, then there exists a continuous function f0 of bounded partial double trigonometric Fourier series of fo diverge over cubes.     

THE FOURIER SERIES EXPANSIONS OF FUNCTIONS DEFINED ON S-SETS

ＴＨＥＦＯＵＲＩＥＲＳＥＲＩＥＳＥＸＰＡＮＳＩＯＮＳＯＦＦＵＮＣＴＩＯＮＳＤＥＦＩＮＥＤＯＮＳＳＥＴＳＬＩＡＮＧＪＩＮＲＯＮＧＬＩＷＡＮＳＨＥＳＵＦＥＮＧＲＥＮＦＵＹＡＯＡｂｓｔｒａｃｔＬｅｔＥｂｅａｃｏｍｐａｃｔｓｓｅｔｓｏｆＲｎ．Ｔｈ...     

THE APPROXIMATION OF CONTINUOUS FUNCTIONS BY PARTIAL SUMS OF ITS FOURIER SERIES

and the factor In n cannot be omitted in gerneral unless f(x) belongs to more smoother class. For the saks of omission of the factor In n, R. Salem and A Zygmund introduced the conception of function to be monotonic type, i.e., for f(x) ∈C_π, if there exists a constant C such that f(x) Cx is monotonic in(-∞, ∞). They proved that:     

ON THE LOCALIZATION AND CONVERGENCE OF MULTIPLE FOURIER INTEGRAL BY BOCHNER-RIESZ MEANS

In this paper we consider lim _(R-) B_R~(f,x_0), in one case that f_x_0 (t) is a ABMV function on [0, ∞], andin another case that f∈L_(m-1)~1(R~) and x~k/~kf∈BV(R) when |k| = m-1 and f(x) = 0 when |x-x_0|<δ for some δ>0. Our theormes improve the results of Pan Wenjie ([1]).     

The HAUSDORFF DIMENSION AND MEASURE OF THE GENERALIZED MORAN FRACTALS AND FOURIER SERIES

ＴｈｅＨＡＵＳＤＯＲＦＦＤＩＭＥＮＳＩＯＮＡＮＤＭＥＡＳＵＲＥＯＦＴＨＥＧＥＮＥＲＡＬＩＺＥＤＭＯＲＡＮＦＲＡＣＴＡＬＳＡＮＤＦＯＵＲＩＥＲＳＥＲＩＥＳ￥ＲＥＮＦＵＴｈＯ；ＬＩＡＮＧＪＩＮＲＯＮＧＡｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｓｔｕｄｉｅｓｔｈ...     

APPROXIMATION OF BOCHNER-RIESZ MEANS 0F CONJUGATE FOURIER INTEGRALS BEL0W THE CRITICAL INDEX

In this paper we study the approximation on set of full measure for functions in Sobolev spacesL_m~1 (R~n) (m∈N) by Bochner-Riesz means of conjugale Fourier integrals below the critical index. Atheorem concerning the precise approximation orders with relation to the number m of space L_m~1 (R~n) andthe index of Bochner-Riesz means is obtained.     

BANACH SPACES AND INEQUALITIES ASSOCIATED WITH NEW GENERALIZATION OF CESàRO MATRIX

Let the triangle matrix A^ru be a generalization of the Cesàro matrix and U ∈{c₀,c,l_∞}.In this study,we essentially deal with the space U(A^ru) defined by the domain of A^ru in the space U and give the bases,and determine the K?the-Toeplitz,generalized K?theToeplitz and bounded-duals of the space U(A^ru).We characterize the classes(l_∞(A^ru):l_∞),(l_∞(A^ru):c),(c(A^ru):c),and...     

THE SERIES SOLUTION OF THE GEL'FAND-LEVITAN-MARCHENKO EQUATION

The convergence of the Neumann series solution of the Gel'fand-Levitan-Marchenko equation in L~∞[0, ∞), L~1[0, ∞) and L~2[0, ∞) is proved and the explicit expression of the series solution is given.     

APPROXIMATION OF FOURIER INTERGRAL BY BOCHNER-RIESZ MEANS UNDER ONE-SIDE CONDITION

In this paper we study the approximation of Fourier integral by Bochner-Riesz means under one-sidecondition and improve the results in[1].     

EXTENDED CESRO OPERATORS ON THE BLOCH SPACE IN THE UNIT BALL OF C~n

The paper defines an extended Cesaro operator Tg with holomorphic symbol g in the unit ball B of Cn asWhere is the radial derivative of g. In this paper, the author charac-     

SYNCHRONIZATION STABILITY OF COMPLEX CONNECTED NETWORKS WITH DELAYS

Recently, much work has been devoted to the study of a large-scale complex system described by a network or a graph with complex topology, whose nodes are the elements of the system and whose edges represent the interactions among them. On the other hand, realistic modelling of many large networks with nonlocal interaction inevitably requires connection delays to be taken into account, since they naturally arise as a consequence of finite information transmission and processing speeds among the units. This paper gives the sufficient conditions guaranteeing the local and global synchronization stability of the complex connected networks by using Lyapunov functional.     

AN ANALOGUE OF A THEOREM OF FERENC LUKÁCS FOR DOUBLE WALSH—FOURIER SERIES

A theorem of Ferenc Lukács states that if a periodic function is integrable in Lebesgue"s sense and has a discontinuity of first kind at some point , then the th partial sum of the conjugate series to its trigonometric Fourier series at divided by converges to as . An analogue of this theorem for Walsh–Fourier series was proved by Rafat Riad. The main aim of the present paper is to extend the latter result from single to double Wals–Fourier series. We consider also functions of two variables which are of bounded variation over a rectangle in the sense of Hardy and Krause. Among others, we present a proof of the existence of the so-called sector limits of such functions at each point.     

MULTIVARIATE FOURIER SERIES OVER A CLASS OF NON TENSOR－PRODUCT PARTITION DOMAINS

This paper finds a way to extend the well-known Fourier methods,to so-called n 1 directions partition domains in n-dimension.In particular,in 2-D and 3-D cases,we study Fourier methods over 3-direction parallel hexagon partitions and 4-direction parallel paralleogram dodecahedron partitions,respectively.It has pointed that,the most concepts and results of Fourier methods on tensor-product case,such as periodicity,orthogonality of Fourier basis system,Partial sum of Fourier series and its approximation behavior,can be moved on the new non tensor-product partiton case.     

ON THE BEST APPROXIMATION MATRIX PROBLEM AND MATRIX FOURIER SERIES

In this paper the concept of positive definite bilinear matrix moment functional. acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b], is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Kiemann -Lebesgue matrix property and the bessel-parseval matrix inequality are proved. The concept of total set vjith respect to a positive definite matrix functional is introduced , and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional, is established.     

APPROXIMATION PROPERTIES OF (C, α) MEANS OF DOUBLE WALSH-FOURIER SERIES

We study the rate of Zp approximation by Cesaro means of the quadratic partial sums of double Walsh-Fourier series offu nctions from Lp.     

APPROXIMATION PROPERTIES OF （C, α） MEANS OF DOUBLE WALSH-FOURIER SERIES

We study the rate of Lp approximation by Cesaro means of the quadratic partial sums of double Walsh-Fourier series of functions from Lp.     

INVARIANCE PRINCIPLE FOR THE INTEGRAL PERIODOGRAM OF A MULTIPLE TIME SERIES

Many authers have studied the asymptotic behavior of the integral periodogram of a Gaussian stationary series. In this paper, the invariance principle for the integral periodogram of a multiple time series is investigated. We have obtained the following theorem:Let X, be a real M -dimensional stationary time series with spectral density matrix f(λ) which satis-fies the conditions:M,and X, can be represented in theform: Xn =whereξ, is the M-dimensional fourth order moment existedstationary martingale series, thenweakly converges to a Gauss process η(λ) withEη(λ) = 0,where V and Q are the second and fourth moment of the martingale seriesξ , respectively.From this theorem, two corollaries have been given, which cover some results of [2][5][6].