6 problems

Let $p\geq 2$ and $d\geq 2$. Let $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ be the square torus, and let $e_N$ be an eigenfunction of $\sqrt{-\Delta}$ with eigenvalue $2\pi N$. The d…

Let $d\geq 3$ be a natural number, let $m$ be a natural number, let $(\mathfrak{a}_{\bm{n}})_{\bm{n}\in S_{d,m}}$ be complex numbers, let $\epsilon>0$, and let…

Let $\phi\in\mathbb{Z}[n]$ be a polynomial of degree $k\geq3$, and let $a\in\ell^2(\mathbb{Z})$. Define the extension operator … where $\alpha\in\mathbb{T}$ and $e(t)=e^{2\pi i t}$…

Let $\phi_1,\phi_2\in\mathbb{Z}[n]$ be polynomials with respective degrees $k_1$ and $k_2$, and define … where $a\in\ell^2(\mathbb{Z})$ and $(\alpha_1,\alpha_2)\in\mathbb{T}^2$. As…

Let $N\in\mathbb{N}$ and let $a\in\ell^2(\mathbb{Z})$. Define the truncated extension operator … where $(\alpha,\beta)\in\mathbb{T}^2$ and $e(t)=e^{2\pi i t}$. Discrete restriction…

Let $n \geq 3$, let $\lambda \geq 1$, and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. For a function $f$ on $\mathbb{T}^n$, write $\widehat f(k)$ for its Fourier coeffi…