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 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.     

A specialization theorem for analytic functions on compact sets

We show that analytic functions defined on a neighborhood of a compact connected subset K of the complex plane can be specialized to algebraic functions defined on a neighborhood of K.     

On the mean-value theorem for analytic functions

A version of the mean-value theorem (formulas of finite increments) for analytic functions is proved. Volyn University, Lutsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1143–1147, August, 1997.     

A real analytic constructive proof of the lojasiewicz theorem

In this paper we eliminate the complex analytic part from original Lojasiewicz's proof and show a constructive method of producing a saucissonnage for a set of polynomials.     

A distortion theorem for the class of typically real functions

The paper continues the studies of the well-known class T of typically real functions f(z) in the disk U = {z:|z| < 1}. The region of values of the system {f(z ₀), f(z ₀), f(r ₁), f(r ₂),…, f(r _n )} in the class T is investigated. Here, z ₀ ∈ U, Im z ₀ ≠ 0, 0 < r _j  < 1 for j = 1,…, n, n ≥ 2. As a corollary, the region of values of f′(z ₀) in the class of functions f ∈ T with fixed values f(z ₀) and f(r _j ) (j = 1,…, n) is determined. The proof is based on the criterion of solvability of the power problem of moments. Bibliography: 10 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 33–45.     

Counterexample to a uniqueness theorem for analytic operator functions

It is proved that there exists a bounded holomorphic operator-function z→F(z), ¦z¦<1, with compact values (in a separable Hilbert space) and such that its boundary values F(S), ¦S¦=1 are compact on one (given) arc of the circle and not compact on the other. The corresponding example is constructed with the help of vectorial Hankel operators.     

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 .

     

Numerical evaluation of analytic functions by Cauchy's theorem

The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.