CONVERGENCE OF DOUBLE WALSH-FOURIER SERIES AND HARDY SPACES

It is proved that the maximal operator of the Marczinkiewicz-Fejér meams of a double Walsh-Fourier series is bounded from the two-dimensional dyadic martingale Hardy space H p to L p (2/3<p<∞) and is of weak type (1,1). As a consequence we obtain that the Marczinkiewicz-Fejér means of a function f∈L 1 converge a.e. to the function in question. Moreover, we prove that these means are uniformly bounded on H p whenever 2/3<p<∞. Thus, in case f∈H p , the Marczinkiewicz-Fejér means conv f in H p norm. The same results are proved for the conjugate means, too.

Estimation of Winds at Different Isobaric Levels Based on the Observed Winds at 850 hPa Level Using Double Fourier Series

A technique based on the double Fourier series is developed to estimate the winds at different isobaric levels forthe limited area domain, 35°E to 140°E and 30°S to 40°N, using the observed winds at 850 hPa lcvcl for the month ofJune. For this purpose the wind field at a level under consideration is taken in the ratio form with that of 850 hPa level and the coefficients of the double Fouricr series are computed. These coefficients are subsequently used to computethe winds which are compared with the actual winds. The results of the double Fourier series technique are comparedwith those of the polynomial surface fitting method developed by Bavadekar and Khaladkar (1 992). The technique isalso applied for the daily wind data of 11. June, 1979 and the validation of the technique is tested for a few radiosondestations of india. The computed winds for these radiosonde stations arc quite close to observed winds.

Synthesis and crystal structure of a novel double 1:11 series arsenotungstate: [Cu^I(phen)_2]_5H_6[Sm(AsW_(11)O_(39))_2]·6H_2O

A novel double 1：11 series arsenotungstate, [Cu^I（phen）2]5H6[Sm（AsW11O39）2]·6H2O （phen=1, 10-phenanthroline） was synthesized by the hydrothermal method and characterized by elemental analysis, 1R spectra, and single-crystal X-ray diffraction. The title compound was monoclinic, space group P2 1/c, a=3.75710（2） nm, b=1.39776（8） nm, c=-3.33735（18） nm, β=96.967（10）°, V=17.3969（17） nm^3, Z=4. Single crystal X-ray structural analyses revealed that the dimeric polyanion, [SmAs2W22O78]^1-, consisted of one central Sm^3＋ ion and two tetradentate heteropoly ligands [ASW11O39]^7-. The Sm^3＋ ion and two tetradentate defective [AsW11O39]^7- anions were joined together by the sharing of four oxygen atoms from each heteropoly ligand. The bond valence sum （BVS） calculations of the title compound suggested that all four Cu atoms were in the ＋1 oxidation sate.

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.

Domain of Euler Mean in the Space of Absolutely p-Summable Double Sequences with 0

In this study, as the domain of four dimensional Euler mean E(r,s) of orders r,sin the space L_p for 0 < p < 1, we examine the double sequence space ε_p^(r,s) and some properties of four dimensional Euler mean. We determine the α-and β(bp)-duals of the space εp r,s, and characterize the classes(ε_p^(r,s):M_u),(ε_p^(r,s):C_(bp)) and(ε_p^(r,s):L_q) of four dimensional matrix transformations, where 1 ≤q < ∞. Finally, we shortly emphasize on the Euler spaces of single and double sequences, and note some further suggestions.

Double sampling derivatives and truncation error estimates

This paper investigates double sampling series derivatives for bivariate functions defined on R2 that are in the Bernstein space. For this sampling series, we estimate some of the pointwise and uniform bounds when the function satisfies some decay conditions. The truncated series of this formula allow us to approximate any order of partial derivatives for function from Bernstein space using only a finite number of samples from the function itself. This sampling formula will be useful in the approximation theory and its applications, especially after having the truncation error well-established. Examples with tables and figures are given at the end of the paper to illustrate the advantages of this formula.

Some Improvement on Hilbert's Inequality and Its Application

A new improvement of Hilbert's inequality for double series can be establishedby means of a strengthened Cauchy's inequality. As application, a quite sharp result onFejer-Riesz's inequality is obtained.

The Influence of Edge Restraining Stiffness on the Transverse Vibrations of Rectangular Plate Structures

This paper presents an analytical study on the influence of edge restraining stiffness on the transverse vibrations of rectangular plate structure. An improved Fourier series method was employed to analyze the transverse vibration of plate structure with general elastically restrained boundary conditions. A linear combination of a double Fourier series and eight auxiliary terms was sought as the admissible function of the flexural displacement of the plate, each term being a combination of a polynomial function and a single cosine series expansion. The auxiliary terms were introduced to ensure and improve the smoothness of the original displacement function and its derivatives at the boundaries. Several numerical examples were given to demonstrate the validity and accuracy of the current solution. The influences of translational and rotational stiffness on the natural frequencies and mode shapes of plate were analyzed by numerical results. The results show that the translational stiffness has bigger influence on the natural frequencies than the rotational stiffness. It is generally well known that little change of the rotational stiffness has little influence on the mode shapes of plate. However, the current work shows that a very little change of rotational stiffness value may lead to a large change of the mode shapes of a square plate structure.

One-dimensional dynamic equations of a piezoelectric semiconductor beam with a rectangular cross section and their application in static and dynamic characteristic analysis

Within the framework of continuum mechanics, the double power series ex- pansion technique is proposed, and a series of reduced one-dimensional （1D） equations for a piezoelectric semiconductor beam are obtained. These derived equations are universal, in which extension, flexure, and shear deformations are all included, and can be degen- erated to a number of special cases, e.g., extensional motion, coupled extensional and flexural motion with shear deformations, and elementary flexural motion without shear deformations. As a typical application, the extensional motion of a ZnO beam is analyzed sequentially. It is revealed that semi-conduction has a great effect on the performance of the piezoelectric semiconductor beam, including static deformations and dynamic be- haviors. A larger initial carrier density will evidently lead to a lower resonant frequency and a smaller displacement response, which is a little similar to the dissipative effect. Both the derived approximate equations and the corresponding qualitative analysis are general and widely applicable, which can clearly interpret the inner physical mechanism of the semiconductor in the piezoelectrics and provide theoretical guidance for further experimental design.

A MULTIPLE q-EXPONENTIAL DIFFERENTIAL OPERATIONAL IDENTITY

Using Hartogs’fundamental theorem for analytic functions in several complex variables and q-partial differential equations,we establish a multiple q-exponential differential formula for analytic functions in several variables.With this identity,we give new proofs of a variety of important classical formulas including Bailey’s 6ψ6 series summation formula and the Atakishiyev integral.A new transformation formula for a double q-series with several interesting special cases is given.A new transformation formula for a 3ψ3 series is proved.

On a Linear Summation Problem of the Double Fourier Series

It this paper we construct a double Fourier series with a new linear summation factor, for the arbitrary continuous periodic function f(x,y)with period 2л, it converges to the function(fx,y) uniformly on total oplane,and its convergence order is the best one.

A Survey of Various Refinements and Generalizations of Hilbert's Inequalities

Expounded in this survey article is a series of refinements and generalizations of Hilbert's inequalities mostly published during the years 1990 through 2002.Those inequalities concerned may be classified into several types (discrete and integral etc.), and various related results obtained respectively by L. C. Hsu, M. Z. Gao, B. C. Yang, J. C. Kuang, Hu Ke and H. Hong et.al are described a little more precisely. Moreover, earlier and recent extensions of Hilbert-type inequalities are also stated for reference. And the new trend and the research ways are also brought forward.

A Refinement of the Weighted Hilbert Inequality

In this paper it is shown that a refinement on the weighted Hilbert inequalityfor double series can be established by introducing a proper non-zero real number R_ω.The expression of R_ω is given by means of the positive definiteness of a Gram matrix.

Numerical Simulation of Thermal Characteristics in the Process of Electroslag Remelting for Production of Large Slab Ingots

The slag pool is a complex system which gathers electromagnetic field,thermal field and flow field in the process of electroslag remelting(ESR)for production of large slab ingots.In this manuscript,mathematic foundation and boundary conditions of the numerical simulation for thermal field in the ESR process of large slab ingots were analyzed, and mathematic model of heat generation in the slag pool and the solidification in the metal molten pool were founded by using the finite element software ANSYS.According to the simulation results,it can be found that the temperature distribution in the process of ESR for production of large slab ingots with double electrode series is different from that in the electroslag furnace with a single electrode.The region of the biggest current density and the highest temperature in the electroslag furnace with a single electrode is below the electrode,while the same region in the process of ESR with the double electrode series for production of large slab ingots locates between the two electrodes.The depth of the metal pool and the temperature of the slag bath simulated by mathematical model were close to the measured value in the experimental process,which verifies the reliability of the simulation method and the model,and it will provide a theoretical basis for the quality control of large slab ingots in the process of ESR.

Singular Integrals with Bilinear Phases

We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by ｜y′｜ 〉 ｜x′｜, and presenting phases λ（Ax ＋ By） with 0≤ A, B ≤ 1 and λ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A1 B and A involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.