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.

POINTWISE CONVERGENCE OF FOURIER-JACOBI SERIES

The purpose of this paper is to study the pointwise and almost everywhere convergence of the Cesaro means (C,δ) of Fourier-Jacobi expansions, the main term of the Lebesgue constant of the (C ,8) means for - 1<δ≤ α+1/2 is obtained. With the aid of the generalized translation in terms of Jacobi polynomials, pointwise convergence theorems of the (C,δ) means for δ>α+1/2 and equiconvergence theorems for - 1<δ≤α+1/2 are proved. The analogues of the Lebesgue, Salem and Young theorems of the Cesaro means at the critical index δ = α+1/2 are established.

ASYMPTOTIC EXPRESSION NEAR THE ELLIPSE OF CONVERGENCE OF JACOBI SERIES IN COMPLEX PLANE

In this paper, we shall give an Abel type theorem of Jacobi series and then based on it discuss asymptotic expressions near the ellipse of convergence of Jacobi series in complex plane.

CHARACTERIZATION OF THE CONVERGENCE DOMAINS OF POLYNOMIAL SERIES AND THE MINIMAL CONVERGENCE DOMAIN

A characterization of the convergence domains of polynomial series is disucssed. the minimal convergence domain for a kind of polynomial series is shown.

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.

Truncated series solutions to the(2+1)-dimensional perturbed Boussinesq equation by using the approximate symmetry method

In this paper, the（2＋1）-dimensional perturbed Boussinesq equation is transformed into a series of two-dimensional（2 D） similarity reduction equations by using the approximate symmetry method. A step-by-step procedure is used to acquire Jacobi elliptic function solutions to these similarity equations, which generate the truncated series solutions to the original perturbed Boussinesq equation. Aside from some singular area, the series solutions are convergent when the perturbation parameter is diminished.

COMPARISONS OF ABSOLUTELY CONVERGENT TRIGONOMETRIC SERIES

In this paper, we first discuss the methods of comparing two special absolutely convergentsine series, sinnx and sinnx. We state the theorem in.one dimensional case as follows; Theorem. Let be convergent series with nonnegative terms. SupposeThen for all x∈[0,π]If, in addition, then

A Novel Method for Determining Microbial Kinetics

Understanding microbial growth is essential to any research conducted in the fields of microbiology and biotechnology. Current methods of determining growth characteristics of microbes involve subjective graphical interpretations of linearized logarithmic data. Reducing error in logistical data decreases disparity between graphical and analytical predictions of microbial characteristics. In this study, a method has been developed to calculate the kinetics of microbial characteristics utilizing a modified Maclaurin series. Convergence of this series approaches the true kinetic value of microbial characteristics to include specific growth rates. In this research, a modified Maclaurin series is used to evaluate microbial kinetics in comparison to graphical determinations.

ON A GENERALIZED TAYLOR THEOREM： A RATIONAL PROOF OF THE VALIDITY OF THE HOMOTOPY ANALYSIS METHOD

A generalized Taylor series of a complex function was derived and some related theorems about its convergence region were given. The generalized Taylor theorem can be applied to greatly enlarge convergence regions of approximation series given by other traditional techniques. The rigorous proof of the generalized Taylor theorem also provides us with a rational base of the validity of a new kind of powerful analytic technique for nonlinear problems, namely the homotopy analysis method.

Quantization of Particle Energy in the Analysis of the Boltzmann Distribution and Entropy

The Boltzmann equilibrium distribution is an important rigorous tool for determining entropy, since this function cannot be measured, but only calculated in accordance with Boltzmann＇s law. On the basis of the commensuration coefficient of discrete and continuous similarly-named distributions developed by the authors, the article analyses the statistical sum in the Boltzmann distribution to the commensuration with the improper integral of the similarly-named function in the full range of the term of series of the statistical sum at the different combination of the temperature and the step of variation （quantum） of the particle energy. The convergence of series based on the Cauchy, Maclaurin criteria and the equal commensuration of series and improper integral of the similarly-named function in each unit interval of variation of series and similarly-named function were estab- lished. The obtained formulas for the commensuration coefficient and statistical sum were analyzed, and a general expres- sion for the total and residual statistical sums, which can be calculated with any given accuracy, is found. Given a direct calculation formula for the Boltzmann distribution, taking into account the values of the improper integral and commensuration coefficient. To determine the entropy from the new expression for the Boltzmann distribution in the form of a series, the conver- gence of the similarly-named improper integral is established. However, the commensuration coefficient of integral and series in each unit interval turns out to be dependent on the number of the term of series and therefore cannot be used to determine the sum of series through the improper integral. In this case, the entropy can be calculated with a given accuracy with a corresponding quantity of the term of series n at a fixed value of the statistical sum. The given accuracy of the statistical sum turns out to be mathematically identical to the fraction of particles with an energy exceeding a given level of the energy barrier equal to the activation energy in the Arrhenius equation. The prospect of development of the proposed method for expressing the Boltzmann distribution and entropy is to establish the relationship between the magnitude of the energy quantum Ae and the properties of the system-forming particles.

New Rapidly Convergent Series Concerning ζ(2k+1)

Values of new series sum（（（2n-1）!ζ（2n））/（2n ＋ 2k）!）α2n from n=1 to ∞,sum（（（2n-1）!ζ（2n））/（2n＋2k ＋1）!）β2n from n=1 to ∞ are given concerning ζ（2k ＋ 1）,where k is a positive integer,α can be taken as 1,1/2,1/3,2/3,1/4,3/4,1/6,5/6 and β can be taken as 1,1/2.Some previous results are included as special cases in the present paper and new series converges more rapidly than those exsiting results for α = 1/3,or α = 1/4,or α = 1/6.

Summing Boolean Algebras

In this paper we will study some families and subalgebras■of■(N)that let us character- ize the unconditional convergence of series through the weak convergence of subseries ∑_(i∈A)x_i,A∈(?). As a consequence,we obtain a new version of the Orlicz Pettis theorem,for Banach spaces.We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.

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.