Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

Zeros of the sum of polynomials

It is given an upper bound for the number of simple and distinct zeros of the polynomial f+g, where f and g are relatively prime polynomials with complex coefficients.     

Approximation of several dimensional functions by trigonometric polynomials

Let f(x, y) be a periodic function defined on the region D

Relaxation results for functions depending on polynomials changing sign on rank-one matrices

In this paper, we are interested in computing the different convex envelopes of functions depending on polynomials, especially those having it is main part change sign on rank-one matrices. Our main result applies to functions of the type W(F)=φ(P(F)), W(F)=φ(P(F))+f(detF) or W(F)=φ(P(F))+g(adjnF) defined on the space of matrices, where φ, f:R→R and g:R³→R are three continuous functions, and P=P₀+P₁+?+Pd is a polynomial such that Pd has the property of changing sign on rank-one matrices. Then the polyconvex, quasi-convex and rank-one convex envelopes of W are equal.     

Some integral and discrete inequalities in several independent variables

We improve the constants in some integral and discrete inequalities in n independent variables which are due to Agarwal and Sheng [1] and Agarwal and Pang [2].     

多复变数的二阶微分从属

在文[1－2]的基础上讨论了多复变数中的二阶微分从属,取得了一些结果。这些结果是文[1-5]中的一些结果的推广。     

A theorem concerning a product of a general class of polynomials and theH-function of several complex variables

A theorem concerning a product of a general class of polynomials and theH-function of several complex variables is given. Using this theorem certain integrals and expansion formula have been obtained. This general theorem is capable of giving a number of new, interesting and useful integrals, expansion formulae as its special cases.     

The star function for meromorphic functions of several complex variables

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.     

Theory of several paravector variables: Bochner-Martinelli formula and Hartogs theorem

The aim of this article is to extend the theory of several complex variables to the non-commutative realm. Some basic results, such as the Bochner-Martinelli formula, the existence theorem of the solutions to the non-homogeneous Cauchy-Riemann equations, and the Hartogs theorem, are generalized from complex analysis in several variables to Clifford analysis in several paravector variables. In particular, the Bochner-Martinelli formula in several paravector variables unifies the corresponding formulas in the theory of one complex variable, several complex variables, and several quaternionic variables with suitable modifications.     

Multifractal spectrum and generic properties of functions monotone in several variables

We study the singularity (multifractal) spectrum of continuous functions monotone in several variables. We find an upper bound valid for all functions of this type, and we prove that this upper bound is reached for generic functions monotone in several variables. Let be the set of points at which f has a pointwise exponent equal to h. For generic monotone functions f:d[0,1]→R, we have that for all h∈[0,1], and in addition, we obtain that the set is empty as soon as h>1. We also investigate the level set structure of such functions.     

Local approximation of functions with several variables

This paper presents the problem of local approximation of scalar functions with several variables, including points of non-differentiability. The procedure considers that the mapping displays rates of change of power type with respect to linear changes in the coordinate domain, and the exponents are not necessarily integer. The approach provides a formula describing the local variability of scalar fields which contains and generalizes Taylor’s formula of first order. The functions giving the contact are Müntz polynomials. The knowledge of their coefficients and exponents enables the finding of local extremes including cases of non-smoothness. Sufficient conditions for the existence of global maxima and minima of concave-convex functions are obtained as well.     

Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the total degree and the vector degree, are considered. We show that the number of irreducibles can be computed recursively by degree and that the number of relatively prime pairs can be expressed in terms of the number of irreducibles. We also obtain asymptotic formulas for the number of irreducibles and the number of relatively prime pairs. The asymptotic formulas for the number of irreducibles generalize and improve several previous results by Carlitz, Cohen and Bodin.     

On the convergence of certain sequences of rational approximants to meromorphic functions in several variables

In a previous paper, the author introduced a new class of multivariate rational interpolants, which are called Optimal Padé-type Approximants (OPTA). There, for this class of rational interpolants, which extends classical univariate Padé Approximants, a direct extension of the “de Montessus de Ballore's Theorem” for meromorphic functions in several variables is established. In the univariate case, this theorem ensures uniform convergence of a row of Pade Approximants when the denominator degree equals the number of poles (counting multiplicities) in a certain disc. When one overshoots the number of poles when fixing the denominator degree, convergence in measure or capacity has been proved and, under certain additional restrictions, the uniform convergence of a subsequence of the row. The author tackles the latter case and studies its generalization to functions in several variables by using OPTA.     

Irreducibility results for compositions of polynomials in several variables

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.     

A result on common zeros location of orthogonal polynomials in two variables

The aim of this paper is to investigate some general properties of common zeros of orthogonal polynomials in two variables for any given region D⊂R² from a view point of invariant factor. An important result is shown that if X₀ is a common zero of all the orthogonal polynomials of degree k then the intersection of any line passing through X₀ and D is not empty. This result can be used to settle the problem of location of common zeros of orthogonal polynomials in two variables. The main result of the paper can be considered as an extension of the univariate case.     

A relationship between the maximum modulus and Taylor coefficients of entire functions of several complex variables

We present results on the relationship between the growth of the maximum modulus and the decay of Taylor coefficients of entire functions of several variables. The results are obtained by two different methods, the first of which had been proposed earlier by Oskolkov for the one-dimensional case, and the second is based on the use of the Legendre-Jung-Fenchel conjugates of the weight functions. Attention is mainly paid to the characterization of the growth of entire functions with respect to the conjunction of variables; however, some results are obtained for the case in which there is different growth with respect to different variables. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 238–258, August, 1997. Translated by N. K. Kulman     

A note on the modified q-Bernstein polynomials for functions of several variables and their reflections on q-Volkenborn integration

In this paper, we consider the modified q-Bernstein polynomials for functions of several variables on q-Volkenborn integral and investigate some new interesting properties of these polynomials related to q-Stirling numbers, Hermite polynomials and Carlitz’s type q-Bernoulli numbers.     

Maps of several variables of finite total variation. I. Mixed differences and the total variation

Given two points a=(a₁,…,an) and b=(b₁,…,bn) from Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), we study properties of the total variation of f on introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217-222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (n=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (n=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61-71] (n∈N) for real-valued functions of n variables.