A division theorem for real analytic functions

We characterize those homogeneous polynomials P [z₁, ... ,z_d] for which the principal ideal (P) = P · A(^d) is complementedin A(^d) or, equivalently, those which admit a continuous lineardivision operator. The condition is the same as that which characterizes,among the homogeneous polynomials, those which are nonellipticand for which P(D) is surjective in A(^d), and those for whichP(D) admits a continuous linear right inverse in C(^d). It dependsonly on the type of real singularities.     

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 .

     

The parabolic analytic functions and the derivative of real functions

Among the bidimensional hypercomplex-number systems defined as the parabolic (dual) numbers are introduced with the rule α = 0. As well as the functions of a complex variable, the analytic functions of a parabolic variable can be introduced as analytic continuation of the real functions of a real variable. These functions hold the property that the “imaginary” part is linked to the derivative of the “real” part. In this paper we will show how this property allows one to demonstrate in an algebraic way some rules of the differential calculus for the real functions of a real variable.     

A simple proof of the approximation by real analytic Lipschitz functions

A theorem in Azagra et al. (preprint) [1] asserts that on a real separable Banach space with separating polynomial every Lipschitz function can be uniformly approximated by real analytic Lipschitz function with a control over the Lipschitz constant. We give a simple proof of this theorem.     

Approximation of analytic functions and their real part

LetG?C be a finite quasidisk, andf(z) an analytic function inG whose real partu(z)?Ref(z) is continuous on \(\bar G\) . Connections between the approximation properties of the functionsu(z) andf(z) are obtained.     

Global complexification of real analytic globally subanalytic functions

We show that a globally subanalytic function that is real analytic can be extended in a globally subanalytic way to a holomorphic function. We also establish a parametric version of this result. For other important ominimal structures, we show that unary definable real analytic functions can be extended definably to a holomorphic function.     

Convolution operator in the space of real analytic functions

Translated from Matematicheskie Zametki, Vol. 49, No. 3, pp. 57–65, March, 1991.     

On the real exponential field with restricted analytic functions

The model-theoretic structure (ℝ_an, exp) is investigated as a special case of an expansion of the field of reals by certain families ofC ^∞-functions. In particular, we use methods of Wilkie to show that (ℝ_an, exp) is (finitely) model complete and O-minimal. We also prove analytic cell decomposition and the fact that every definable unary function is ultimately bounded by an iterated exponential function. An erratum to this article is available at .     

Note on Schwarz-Pick estimates for bounded and positive real part analytic functions

In this note we consider higher order derivatives of bounded analytic functions.

     

A simple quadrature-type method for the computation of real zeros of analytic functions in finite intervals

A new direct approximate method for the computation of zeros of analytic functions is proposed. For such a function possessing one real zero in a finite part of the real axis, this method permits the determination of this zero with a satisfactory accuracy by using a quite elementary algorithm. The present method is based on the Gauss- and Lobatto-Chebyshev quadrature rules for Cauchy type principal value integrals and is always convergent. The simplicity, accuracy and unrestricted convergence of the proposed method make it competitive to the analogous classical elementary methods. Numerical results are also presented.     

A subanalytic triangulation theorem for real analytic orbifolds

Let X be a real analytic orbifold. Then each stratum of X is a subanalytic subset of X. We show that X has a unique subanalytic triangulation compatible with the strata of X. We also show that every Cr-orbifold, 1?r?∞, has a real analytic structure. This allows us to triangulate differentiable orbifolds. The results generalize the subanalytic triangulation theorems previously known for quotient orbifolds.     

A uniqueness result for harmonic functions

Let , , and suppose is harmonic in and on the closure of . If the gradient of vanishes continuously on a subset of of positive -dimensional Lebesgue measure and satisfies certain regularity conditions, then must be identically constant.

     

Inheritance of surjectivity for partial differential operators on spaces of real analytic functions

For an open set let A(Ω) be the space of real analytic functions on Ω. Improving our previous results, we prove a new quantitative characterization of the linear partial differential operators P(D) which are surjective on A(Ω). This implies that P(D) is surjective on if P(D) is surjective on A(Ω) for some Ω≠∅. Further inheritance properties for the surjectivity of P(D) on A(Ω) are also obtained.     

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.