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Necessary and sufficient conditions are presented for several families of planar curves to form a set of stable sampling for the Bernstein space $mathcal{B}_{Omega}$ over a convex set $Omega subset mathbb{R}^2$. These conditions essentially describe the mobile sampling property of these families for the Paley-Wiener spaces $mathcal{PW}^p_{Omega},1leq p<infty$.
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We discuss sampling constants for dominating sets in Bergman spaces. Our method is based on a Remez-type inequality by Andrievskii and Ruscheweyh. We also comment on extensions of the method to other spaces such as Fock and Paley-Wiener spaces.
We provide a sufficient condition for sets of mobile sampling in terms of the surface density of the set.
In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{omega _varphi}$, where $omega _varphi= e^{-varphi}$ and $varphi$ is a subharmonic function. We consider compact Hankel operators $H_{overline {phi}}$, with anti-analytic symbols ${overline {phi}}$, and give estimates of the trace of $h(|H_{overline phi}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{overline {phi}}))_n$ in terms of decreasing rearrangement of $|phi |/sqrt{Delta varphi}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{overline {phi}}))_n$ is achieved by $(s_n (H_{overline{z}}))_n$. Namely, we establish that if $s_n(H_{overline {phi}})= o (s_n(H_{overline {z}}))$, then $H_{overline {phi}} = 0$. Then, we show that if $Delta varphi (z) asymp frac{1}{(1-|z|^2)^{2+beta}}$ with $beta geq 0$, then $s_n(H_{overline {phi}}) = O(s_n(H_{overline {z}}))$ if and only if $phi $ belongs to the Hardy space $H^p$, where $p= frac{2(1+beta)}{2+beta}$. Finally, we compute the asymptotics of $s_n(H_{overline {phi}})$ whenever $ phi in H^{p }$.
We give in this paper some equivalent definitions of the so called $rho$-Carleson measures when $rho(t)=(log(4/t))^p(loglog(e^4/t))^q$, $0le p,q<infty$. As applications, we characterize the pointwise multipliers on $LMOA(mathbb S^n)$ and from this space to $BMOA(mathbb S^n)$. Boundedness of the Ces`aro type integral operators on $LMOA(mathbb S^n)$ and from $LMOA(mathbb S^n)$ to $BMOA(mathbb S^n)$ is considered as well.
A simple proof of Ramanujans formula for the Fourier transform of the square of the modulus of the Gamma function restricted to a vertical line in the right half-plane is given. The result is extended to vertical lines in the left half-plane by solving an inhomogeneous ODE. We then use it to calculate the jump across the imaginary axis.
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