Réunion d'hiver SMC 2015
Université McGill, 4 - 7 décembre 2015
\item[(b)] $(D^{\alpha}f)(x)=(D^{\alpha}g)(x)$ if $x\in T$. \end{enumerate} We show that if $C\subseteq\mathbb{R}^{n+1}$ is meager, $A\subseteq\mathbb{R}^n$ is countable and for each multi-index $\alpha$ and $p\in A$ we are given a countable dense set $A_{p,\alpha}\subseteq\mathbb{R}$, then we can require also that
\begin{enumerate} \item[(c)] $(D^\alpha f)(p)\in A_{p,\alpha}$ for $p\in A$ and $\alpha$ any multi-index;
\item[(d)] if $x\not\in T$, $q=(D^\alpha f)(x)$ and there are values of $p\in A$ arbitrarily close to $x$ for which $q\in A_{p,\alpha}$, then there are values of $p\in A$ arbitrarily close to $x$ for which $q=(D^\alpha f)(p)$;
\item[(e)] for each $\alpha$, $\{x\in\mathbb{R}^n:(x,(D^\alpha f)(x))\in C\}$ is meager in $\mathbb{R}^n$. \end{enumerate} Clause (d) is a surjectivity property which can be strengthened to allow for finding solutions in $A$ to equations of the form $q=h^*(x,(D^\alpha f)(x))$ under similar assumptions, where $h(x,y)=(x,h^*(x,y))$ is one of countably many given fiber-preserving homeomorphisms of open subsets of $\mathbb{R}^{n+1}\cong\mathbb{R}^n\times\mathbb{R}$.
We also prove a weaker corresponding result with ``meager'' replaced by ``Lebesgue null.'' In this context, the approximating function is $C^\infty$ rather than entire, and we do not know whether it can be taken to be entire.
The Lorentz spaces, $\Lambda^p(v)$, provide a powerful tool to obtain Fourier inequalities in weighted Lebesgue spaces. Our focus is on the Fourier series in weighted Lorentz spaces. We provide relations between weight functions and exponents, that are necessary and sufficient for the boundedness of the Fourier coefficients, viewed as a map between Lorentz spaces. We also apply our results to weighted Lebesgue spaces, $L\log L$, and Lorentz-Zygmund spaces.
This is joint work with Gord Sinnamon.
Fatemeh Sharifi, University of Western ontario,
Title: Zero free approximation.
Abstract. Let $E$ be a compact subset of the complex plane with connected complement. We define $A(E)$ to be the class of all complex continuous functions on $E$ that are holomorphic in the interior $E^0$ of $E$. The remarkable theorem of Mergelyan states that every $f\in A(E)$ is uniformly approximable by polynomials on $E$, but is it possible to realize such an approximation by polynomials that are zero-free on $E$? This question was proposed (but not pubished) by P. Gauthier and subsequently posed independently (and published) by J. Andersson. Recently, Arthur Danielyan described a class of functions for which zero-free approximation is possible on an arbitrary $E$. I intend to present a generalization of his work on Riemann surfaces. This is joint work with Paul Gauthier.