scholarly journals On the Borel mapping in the quasianalytic setting

2017 ◽  
Vol 121 (2) ◽  
pp. 293 ◽  
Author(s):  
Armin Rainer ◽  
Gerhard Schindl
Keyword(s):  
Sequence Space ◽  
Smooth Functions ◽  
The Real ◽  
Partial Derivatives ◽  
Real Analytic ◽  
Borel Mapping ◽  
Analytic Class

The Borel mapping takes germs at $0$ of smooth functions to the sequence of iterated partial derivatives at $0$. We prove that the Borel mapping restricted to the germs of any quasianalytic ultradifferentiable class strictly larger than the real analytic class is never onto the corresponding sequence space.

scholarly journals On the Construction of Large Algebras Not Contained in the Image of the Borel Map

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Céline Esser ◽  
Gerhard Schindl
Keyword(s):  
Sequence Space ◽  
Analytic Functions ◽  
General Setting ◽  
Smooth Functions ◽  
The Real ◽  
Cauchy Product ◽  
Real Analytic Functions ◽  
Borel Map ◽  
Real Analytic ◽  
The Stability

AbstractThe Borel map $$j^{\infty }$$j∞ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $$j^{\infty }$$j∞ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In a recent paper the authors have studied the size of the image of $$j^{\infty }$$j∞ by using different approaches and worked in the general setting of quasianalytic ultradifferentiable classes defined by weight matrices. The aim of this paper is to show that the image of $$j^{\infty }$$j∞ is also small with respect to the notion of algebrability and we treat both the Cauchy product (convolution) and the pointwise product. In particular, a deep study of the stability of the considered spaces under the pointwise product is developed.

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

The Dynamics of a Piecewise Linear Map and its Smooth Approximation

1997 ◽  
Vol 07 (02) ◽  
pp. 351-372 ◽  
Author(s):  
D. Aharonov ◽  
R. L. Devaney ◽  
U. Elias
Keyword(s):  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Invariant Curves ◽  
Smooth Approximation ◽  
Linear Map ◽  
Island Structure ◽  
The Real ◽  
Piecewise Linear Map ◽  
Real Analytic ◽  
Area Preserving

The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fε: (x, y) → (1 - y - fε(x), x), where fε(x) is real analytic and close to |x| for small values of ε. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fε, provided that ε is small.

scholarly journals Arc-smooth functions on closed sets

2019 ◽  
Vol 155 (4) ◽  
pp. 645-680 ◽  
Author(s):  
Armin Rainer
Keyword(s):  
Holomorphic Function ◽  
Smooth Function ◽  
Smooth Curve ◽  
Smooth Functions ◽  
Closed Sets ◽  
Open Set ◽  
Connected Set ◽  
Subanalytic Sets ◽  
Real Analytic ◽  
Analytic Curves

By an influential theorem of Boman, a function $f$ on an open set $U$ in $\mathbb{R}^{d}$ is smooth (${\mathcal{C}}^{\infty }$) if and only if it is arc-smooth, that is, $f\,\circ \,c$ is smooth for every smooth curve $c:\mathbb{R}\rightarrow U$. In this paper we investigate the validity of this result on closed sets. Our main focus is on sets which are the closure of their interior, so-called fat sets. We obtain an analogue of Boman’s theorem on fat closed sets with Hölder boundary and on fat closed subanalytic sets with the property that every boundary point has a basis of neighborhoods each of which intersects the interior in a connected set. If $X\subseteq \mathbb{R}^{d}$ is any such set and $f:X\rightarrow \mathbb{R}$ is arc-smooth, then $f$ extends to a smooth function defined on $\mathbb{R}^{d}$. We also get a version of the Bochnak–Siciak theorem on all closed fat subanalytic sets and all closed sets with Hölder boundary: if $f:X\rightarrow \mathbb{R}$ is the restriction of a smooth function on $\mathbb{R}^{d}$ which is real analytic along all real analytic curves in $X$, then $f$ extends to a holomorphic function on a neighborhood of $X$ in $\mathbb{C}^{d}$. Similar results hold for non-quasianalytic Denjoy–Carleman classes (of Roumieu type). We will also discuss sharpness and applications of these results.

scholarly journals Extension sets for real analytic functions and applications to Radon transforms

1996 ◽  
Vol 19 (4) ◽  
pp. 625-632
Author(s):  
Vernor Arguedas ◽  
Ricardo Estrada
Keyword(s):  
Entire Function ◽  
Analytic Functions ◽  
Radon Transforms ◽  
Support Theorem ◽  
The Real ◽  
Real Analytic Functions ◽  
Small Set ◽  
Real Analytic

The real analytic character of a functionf(x,y)is determined from its behavior along radial directionsfθ(s)=f(scosθ,ssinθ)forθ∈E, whereEis a “small” set. A support theorem for Radon transforms in the plane is proved. In particular iffθextends to an entire function forθ∈Eandf(x,y)is real analytic inℝ2then it also extends to an entire function inℂ2.

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