Infinite systems of linear equations for real analytic functions

We study the problem when an infinite system of linear functional equations

has a real analytic solution on for every right-hand side and give a complete characterization of such sequences of analytic functionals . We also show that every open set has a complex neighbourhood such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on .

     

Integral means and arclength inequalities for typically real logharmonic mappings

In this paper, we use star functions to conclude the integral means inequalities for typically real logharmonic mappings. Moreover, we determine the upper bound for the arclength of typically real logharmonic mappings.     

广义解析函数的带位移的非线性边值问题

考虑多复变广义解析函数的一个带位移的非线性边值问题,首先将其化为奇异积分方程问题,然后利用Schauder不动点原理及压缩映象原理,证明了解的存在性并给出解的积分表达式.     

On the Riemann‐Hilbert boundary value problem for generalized analytic functions in the framework of variable exponent spaces

Let Γ be a simple closed curve that bounds the finite domain D , z =z (ζ )=z (r e ^{i ?} ) be the conformal mapping of the circle {ζ :|ζ |<1} onto the domain D . Furthermore, let the functions A (z ), B (z ) be given on D and U ^{s ,2}(A ;B ;D ) be the set of regular solutions of the equation . We call the Smirnov class E ^{p (t )}(A ;B ;D ) the set of those generalized functions W in D for which where p (t ) is a positive measurable function on Γ. We consider the Riemann‐Hilbert problem: Define a function W (z ) from the class E ^{p (t )}(A ;B ;D ) for which the equality, is fulfilled almost everywhere on Γ. It is assumed that Γ is a piecewise‐smooth curve without external peaks; , p is Log Hölder continuous and the function belongs to the class A (p (t );Γ), which is the generalization of the well‐known Simonenko class A (p ;Γ), where p is constant. The solvability conditions are established, and solutions are constructed.     

A Runge theorem for generalized analytic functions

We show an approximation theorem of Runge type for solutions of the generalized Vekua equation $Lu=Au+B\overline{u}$, where L belongs to a class of degenerate elliptic planar vector fields and $A,B\in L^p$. To prove the theorem, we make use of an integral representation for the solutions of the generalized Vekua equation valid on relatively compact sets. As an application, we study the global solvability of the equation $Lu=Au+B\overline{u}+f$ with $f\in L^p$ and some of its consequences.     

Nonstandard principles for generalized functions

We show that principles from nonstandard analysis hold to some extent for nonlinear generalized functions. The generalized functions under consideration are constructed as families of functions modulo a free filter, as it is usually done in applied analysis. In contrast with models of nonstandard analysis, we do not require the filter to be an ultrafilter. The principles are intended to be used as a tool for proving theorems, which we illustrate by means of an automatic continuity result that was not suspected by experts in the field.     

Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients

We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.

     

A characterization of generalized poles of generalized Nevanlinna functions

Generalized poles of a generalized Nevanlinna function Q ∈ ??_κ (??) are defined in terms of the operator representation of Q . In this paper those generalized poles that are not of positive type and their degrees of non‐positivity are characterized analytically by means of pole cancellation functions. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Merit functions and error bounds for constrained mixed set-valued variational inequalities via generalized f-projection operators

In this paper, we introduce and investigate a constrained mixed set-valued variational inequality (MSVI) in Hilbert spaces. We prove the solution set of the constrained MSVI is a singleton under strict monotonicity. We also propose four merit functions for the constrained MSVI, that is, the natural residual, gap function, regularized gap function and D-gap function. We further use these functions to obtain error bounds, i.e. upper estimates for the distance to solutions of the constrained MSVI under strong monotonicity and Lipschitz continuity. The approach exploited in this paper is based on the generalized f-projection operator due to Wu and Huang, but not the well-known proximal mapping.     

On generalized means and generalized convex functions

Properties of generalized convex functions, defined in terms of the generalized means introduced by Hardy, Littlewood, and Polya, are easily obtained by showing that generalized means and generalized convex functions are in fact ordinary arithmetic means and ordinary convex functions, respectively, defined on linear spaces with suitably chosen operations of addition and multiplication. The results are applied to some problems in statistical decision theory.This research was supported by Project No. NR-047-021, Contract No. N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas, Austin, Texas, and by NSF Grant No. ENG-76-10260 at Northwestern University, Evanston, Illinois.     

Spatial pseudoanalytic functions arising from the factorization of linear second order elliptic operators

Biquaternionic Vekua‐type equations arising from the factorization of linear second order elliptic operators are studied. Some concepts from classical pseudoanalytic function theory are generalized onto the considered spatial case. The derivative and antiderivative of a spatial pseudoanalytic function are introduced and their applications to the second order elliptic equations are considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Salagean算子定义的双单叶函数子类的系数估计

In this paper, a new subclass N_Σ~(h,p)(m, λ, μ) of analytic and bi-univalent functions in the open unit disk U is defined by salagean operator. We obtain coefficients bounds |a_2| and |a_3| for functions of the class. Moreover, we verify Brannan and Clunie's conjecture |a_2| ≤2~(1/2)for some of our classes. The results in this paper extend many results recently researched by many authors.     

Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

Let be an open set and let denote the class of real analytic functions on . It is proved that for every surjective linear partial differential operator and every family depending holomorphically on there is a solution family depending on in the same way such that The result is a consequence of a characterization of Fréchet spaces such that the class of ``weakly' real analytic -valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if is replaced by another set.

     

Strongly internal sets and generalized smooth functions

Based on a refinement of the notion of internal sets in Colombeau's theory, so-called strongly internal sets, we introduce the space of generalized smooth functions, a maximal extension of Colombeau generalized functions. Generalized smooth functions as morphisms between sets of generalized points form a sub-category of the category of topological spaces. In particular, they can be composed unrestrictedly.     

正则的正实部函数的导数估计

讨论了正则的正实部函数的导数估计问题,利用正实部函数的性质,得到三阶导数、四阶导数的准确估计式.