Related papers: The Zero Set of a Real Analytic Function

We prove a conjecture of Kurdyka stating that every arc-symmetric semialgebraic set is precisely the zero locus of an arc-analytic semialgebraic function. This implies, in particular, that arc-symmetric semialgebraic sets are in one-to-one…

We characterize precisely the possible rate of decay of the anti-analytic half of a trigonometric series converging to zero almost everywhere.

We study properties of temperate non-negative purely atomic measures in the Euclidean space such that the distributional Fourier transform of these measures are pure point ones. A connection between these measures and almost periodicity is…

This theorem is based on holomorphy of studied functions and the fact that near a singularity point the real part of some rational function can take an arbitrary preassigned value.

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane

We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence $\left\{\frac{\mathrm{d}^n}{\mathrm{d}z^n}\left(R(z)\exp{T(z)}\right)\right\}$. Here, $R(z)$ is a rational function with at…

The Dirichlet--Hardy space $\Ht$ consists of those Dirichlet series $\sum_n a_n n^{-s}$ for which $\sum_n |a_n|^2<\infty$. It is shown that the Blaschke condition in the half-plane $\operatorname{Re} s>1/2$ is a necessary and sufficient…

Suppose we are given a symmetric operator T acting on a subspace of L2{M,m} where M is a connected manifold and m is a measure positive on open sets. Then there is at most one eigenspace that contains a real valued eigenfunction whose set…

We obtain various general conditions in terms of the balayage and Green's functions under which the sequence of points is the zero set for weighted spaces of holomorphic functions in a domain on the complex plane.

A method of constructing an entire function with given zeros and estimates of growth is suggested. It gives a possibility to describe zero sets of certain classes of entire functions of one and several variables in terms of growth of volume…

We show that, in a generic setting, self-affine and almost self-affine measures are exact dimensional, with local dimension equal almost everywhere to the information dimension and given by the zero of a superadditive pressure functional.

It is shown that every C-semianalytic arc-symmetric set can be realized as the zero locus of an arc-analytic function. As a consequence, a Nash globally subanalytic arc-symmetric set is the zero locus of a continuous globally-subanalytic…

In this note, we give a simple proof that the values of the trigonometric functions at any nonzero rational number are transcendental numbers.

We show that the following are consistent with ZFC: 1. Strongly meager sets form an ideal with the same additivity as the ideal of meager sets. 2. There exists a strong measure zero set of size > d (dominating number).

In this article, various results will be demonstrated that enable the delimitation of a zero-free region for holomorphic functions on a set $K$, studying the behavior of their imaginary or real part on the boundary of $K$. These findings…

We give a general approach to infinite dimensional non-Gaussian Analysis for measures which need not have a logarithmic derivative. This framework also includes the possibility to handle measures of Poisson type.

We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal…

On a real ($\mathbb F=\mathbb R$) or complex ($\mathbb F=\mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\it tracks} $X$ if $[Y,X]=fX$ for some continuous function…

We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the…

In this paper we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected $3$-manifold without boundary is either…