Strobl22

Applied Harmonic Analysis and Friends

June 19th - 25th 2022

Strobl, AUSTRIA

"Discrete Translates in the Hardy Space $H^1(\mathbb{R})$"

Dharra, Bhawna

The Hardy space $H^1(\mathbb{R})$ appears as a good substitute for the Lebesgue space $L^1(\mathbb{R})$ as some fundamental operators in Harmonic Analysis, like singular integral operators, maximal operators, and Littlewood-Paley functions, are well behaved (bounded) on the Lebesgue spaces $L^p(\mathbb{R})$, $1<p<\infty$ and $H^1(\mathbb{R})$ but are not bounded on $L^1(\mathbb{R})$. In this talk, we investigate the completeness of systems formed by discrete translates of a function in $H^1(\mathbb{R})$. For a function $f$ on $\mathbb{R}$ and $\lambda \in \mathbb{R}$, define $\tau_{\lambda}f(x):=f(x-\lambda)$. The classical Wiener-Tauberian theorem characterizes those functions $\phi$ whose all translates are complete in $L^1(\mathbb{R})$ in terms of the zero set of its Fourier transform. To be precise, Wiener proved that $span \{\tau_{\lambda}\phi\}_{\lambda \in \mathbb{R} }$ is dense in $L^1(\mathbb{R})$ if and only if its Fourier transform $\hat{\phi}$ never vanishes. We are particularly interested in system of discrete translates, i.e., when $\Lambda$ is a discrete subset of $\mathbb{R}$. Bruna-Olevskii-Ulanovskii proved that for a discrete set $\Lambda \subset \mathbb{R}$, there exists $f \in L^1(\mathbb{R})$ such that $ \{\tau_{\lambda}f\}_{\lambda \in \Lambda}$ is complete in $ L^1(\mathbb{R})$ if and only if its Beurling-Malliavin density $D_{BM}(\Lambda)$ is not finite. We characterize all the discrete sets $\Lambda \subset \mathbb{R}$ in $H^1(\mathbb{R})$ for which there exists a function $f \in H^1(\mathbb{R})$ whose $\Lambda$-translates are complete in $H^1(\mathbb{R})$. We further showed that when $\Lambda$ is a very small perturbation of integers, there exists a pair of functions such that their $\Lambda$-translates are complete in $H^1(\mathbb{R})$. In conclusion, $H^1(\mathbb{R})$ behaves similar to $L^1(\mathbb{R})$ when we consider properties of system of translates. This talk is based on the results of a joint research with S. Sivananthan.
http://univie.ac.at/projektservice-mathematik/e/talks/Dharra_2022-02_strobl22.pdf

« back