In the paper A problem on completeness of exponentials (Annals of Mathematics 178 (2013), 983-1016), the author A. Poltoratski studies the following problem:

Let $\mu$ be a finite positive measure on $\mathbb{R}.$ For $a>0$, let $$ \mathcal{E}_a = \{ e^{ist} : s \in [0,a] \} $$ be the set of exponentials with frequencies in the interval $[0,a]$. The paper studies the question for which minimal $a>0$ is the linear span of $\mathcal{E}_a$ dense in $L^p(\mu)$.

At the moment I'm trying to make use of a certain density of exponentials in $L^p(\mu)$. In fact I'm serching for results on a weaker form of the above problem:

Let $\mu$ be a finite positive measure on $\mathbb{R}$ and $p \in [1,\infty]$. Which conditions on $\mu$ and $p$ imply that the set of exponentials $$\mathcal{E} = \{ e^{ist} : s \in \mathbb{R} \}$$ is complete $L^p(\mu)$?

In this problem, the values $s$ can be chosen from whole $\mathbb{R}$ and not just from a finite interval. I think major difficulty should be for $p \neq 2$ and $p \neq 1$. Are there papers dealing with this kind of question?

Thanks in advance for any comments.

In the paper A problem on completeness of exponentials (Annals of Mathematics 178 (2013), 983-1016), the author A. Poltoratski studies the following problem:

Let $\mu$ be a finite positive measure on $\mathbb{R}.$ For $a>0$, let $$ \mathcal{E}_a = \{ e^{ist} : s \in [0,a] \} $$ be the set of exponentials with frequencies in the interval $[0,a]$. The paper studies the question for which minimal $a>0$ is the linear span of $\mathcal{E}_a$ dense in $L^p(\mu)$.

At the moment I'm trying to make use of a certain density of exponentials in $L^p(\mu)$. In fact I'm serching for results on a weaker form of the above problem:

Let $\mu$ be a finite positive measure on $\mathbb{R}$ and $p \in [1,\infty]$. Which conditions on $\mu$ and $p$ imply that the set of exponentials $$\mathcal{E} = \{ e^{ist} : s \in \mathbb{R} \}$$ is complete $L^p(\mu)$?

In this problem, the values $s$ can be chosen from whole $\mathbb{R}$ and not just from a finite interval. I think major difficulty should be for $p \neq 2$ and $p \neq 1$. Are there papers dealing with this kind of question?

In the paper A problem on completeness of exponentials (Annals of Mathematics 178 (2013), 983-1016), the author A. Poltoratski studies the following problem:

Let $\mu$ be a finite positive measure on $\mathbb{R}.$ For $a>0$, let $$ \mathcal{E}_a = \{ e^{ist} : s \in [0,a] \} $$ be the set of exponentials with frequencies in the interval $[0,a]$. The paper studies the question for which minimal $a>0$ is the linear span of $\mathcal{E}_a$ dense in $L^p(\mu)$.

At the moment I'm trying to make use of a certain density of exponentials in $L^p(\mu)$. In fact I'm serching for results on a weaker form of the above problem:

Let $\mu$ be a finite positive measure on $\mathbb{R}$. Which conditions on $\mu$ imply that the set of exponentials $$\mathcal{E} = \{ e^{ist} : s \in \mathbb{R} \}$$ is complete $L^p(\mu)$?

In this problem, the values $s$ can be chosen from whole $\mathbb{R}$ and not just from a finite interval. Are there papers dealing with this kind of question?

Thanks in advance for any comments.

In the paper A problem on completeness of exponentials (Annals of Mathematics 178 (2013), 983-1016), the author A. Poltoratski studies the following problem:

Let $\mu$ be a finite positive measure on $\mathbb{R}.$ For $a>0$, let $$ \mathcal{E}_a = \{ e^{ist} : s \in [0,a] \} $$ be the set of exponentials with frequencies in the interval $[0,a]$. The paper studies the question for which minimal $a>0$ is the linear span of $\mathcal{E}_a$ dense in $L^p(\mu)$.

At the moment I'm trying to make use of a certain density of exponentials in $L^p(\mu)$. In fact I'm serching for results on a weaker form of the above problem:

Let $\mu$ be a finite positive measure on $\mathbb{R}$ and $p \in [1,\infty]$. Which conditions on $\mu$ and $p$ imply that the set of exponentials $$\mathcal{E} = \{ e^{ist} : s \in \mathbb{R} \}$$ is complete $L^p(\mu)$?

In this problem, the values $s$ can be chosen from whole $\mathbb{R}$ and not just from a finite interval. I think major difficulty should be for $p \neq 2$ and $p \neq 1$. Are there papers dealing with this kind of question?

Thanks in advance for any comments.