arXiv subjects

K. Makridis

Publications and source records attributed to K. Makridis.

4 recordsLinked to original sources

arXiv2017

Nowhere differentiable functions with respect to the position

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 $\partial Ω$ with respect to the position. We give specific examples where each case of the previous dichotomy holds. We also extend the previous result to unbounded domains.

math.CV↗

arXiv2015

Simultaneous Universal Padé Approximation

We prove simultaneous universal Padé approximation for several universal Padé approximants of several types. Our results are generic in the space of holomorphic functions, in the space of formal power series as well as in a subspace of $A^{\infty}$. These results are valid for one center of expansion or for several centers as well.

math.CV↗

arXiv2015

Simultaneous Universal Pade-Taylor Approximation

We prove simultaneous Universal Approximation of a certain type of Pade Approximants and of Taylor series with the same indexes. This is a generic phenomenon in the space of holomorphic functions in any simply connected domain, as well as in several other spaces. Our results are valid for one center of expansion and for several centers, as well.

math.CV↗

arXiv2013

Sets of uniqueness for uniform limits of polynomials in several complex variables

We investigate the sets of uniform limits $A(\bar{B}_n)$, $A(\bar{D}^I)$ of polynomials on the closed unit ball $\bar{B}_n$ of $\mathbb{C}^n$ and on the cartesian product $\bar{D}^I$ where $I$ is an arbitrary set and $\bar{D}$ is the closed unit disc in $\mathbb{C}$. We introduce the notion of set of uniqueness for $A(\bar{D}^I)$ (respectively for $A(\bar{B}_n)$) for compact subsets $K$ of $T^I$ (respectively of $\partial \bar{B}_n$) where $T=\partial D$ is the unit circle. Our main result is that if $K$ has positive measure then $K$ is a set of uniqueness. The converse does not hold. Finally, we do a similar study when the uniform convergence is not meant with respect to the usual Euclidean metric in $\mathbb{C}$, but with respect to the chordal metric $χ$ on $\mathbb{C} \cup \{\infty \}$.

math.CV↗