I. Kh. Musin, P. V. Yakovleva
Description of linear continuous functionals on a space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in $\mathbb R^n$ in terms of their Fourier-Laplace transform is obtained.
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We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.
We construct examples of twice differentiable functions in $\mathbb{R}^n$ with continuous Laplacian and unbounded Hessian. The same construction is also applicable to higher order differentiability.
The problem of representation of elements of weighted space of infinitely differentiable functions on real line by exponential series is considered.
The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result for description of strong dual for weighted spaces of infinitely differentiable functions on real line and weighted spaces of sequences of infinitely differentiable functions on real line in terms of the Fourier-Laplace transform of functionals.
A space $G(M, \varPhi)$ of infinitely differentiable functions in ${\mathbb R}^n$ constructed with a help of a family $\varPhi=\{φ_m\}_{m=1}^{\infty}$ of real-valued functions $φ_m \in~C({\mathbb R}^n)$ and a logarithmically convex sequence $M$ of positive numbers is considered in the article. In view of conditions on $M$ each function of $G(M, \varPhi)$ can be extended to an entire function in ${\mathbb C}^n$. Imposed conditions on $M$ and $\varPhi$ allow to describe the space of such extensions.
Quasianalytic classes are classes of infinitely differentiable functions that satisfy the analytic continuation property enjoyed by analytic functions. Two general examples are quasianalytic Denjoy-Carleman classes (of origin in the analysis of linear partial differential equations) and the class of infinitely differentiable functions that are definable in a polynomially bounded o-minimal structure (of origin in model theory). We prove a generalization to quasianalytic functions of Malgrange's celebrated theorem on the division of infinitely differentiable by real-analytic functions.
For each positive integer k, we describe a map f from the complex plane to a suitable non-complete complex locally convex space such that f is k times continuously complex differentiable but not k+1 times, and hence not complex analytic. We also describe a complex analytic map from l^1 to a suitable complete complex locally convex space which is unbounded on each non-empty open subset of l^1. Furthermore, we present a smooth map from the real line to a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point. As a byproduc…
We investigate the connectedness properties of the set $ I^{\!+\!}(f) $ of points where the iterates of an entire function $ f $ are unbounded. In particular, we show that $ I^{\!+\!}(f) $ is connected whenever iterates of the minimum modulus of $ f $ tend to infinity. For a general transcendental entire function $ f $, we show that $ I^{\!+\!}(f) \cup \lbrace \infty \rbrace $ is always connected and that, if $ I^{\!+\!}(f) $ is disconnected, then it has uncountably many components, infinitely many of which are unbounded.
Let $Ω$ be a bounded domain in $\mathbb{C}$ such that $\partial Ω$ does not contain isolated points. Let $R(Ω)$ be the space of uniform limits on $\overlineΩ$ of rational functions with poles off $\overlineΩ$, endowed with the supremum norm. We prove that either generically all functions $f$ in $R(Ω)$ satisfy % $$ \limsup_{\substack{z \to z_0 z \in \partial Ω}} \Big| \frac{f(z) - f(z_0)}{z - z_0} \Big| = + \infty $$ for every $z_0 \in \partial Ω$ or no such function in $R(Ω)$ meets this requirement. In the first case, the generic function $f \in R(Ω)$ is nowhere differentiable on $\partial Ω$ …
Consider a complex line bundle over a compact complex manifold equipped with an infinitely differentiable metric with strictly positive curvature form. Assign to positive tensor powers of this bundle the associated product metrics and Bergman projection operators. Zelditch has conjectured that if the Bergman kernels, away from the diagonal, decay exponentially fast to zero as the power tends to infinity, then the metric must be real analytic. Moreover, this is conjectured even if exponential decay is assumed to hold merely for some arbitrarily sparse subsequence of powers tending to infinity. …
The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a countable, locally finite collection of arbitrary sets with the sign +1 or -1 prescribed such that the closures of opposite-signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in an Euclidean space and closed condensers of finitely many plates). Theref…
Sum of the exponential and a polynomial: Singular values and Baker wandering domains
本文研究形如指数函数复合与多项式之和的整函数，证明其无穷远点为Baker遗漏值，具有无限临界值且无有限渐近值，并证明此类函数无Baker游荡域，为相关猜想提供候选研究对象。
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