19 problems

Optimal sampling-rate conjecture. One may conjecture that

Chowla's conjecture. There is an absolute constant $c>0$ such that, uniformly over all sets $A$ of $N$ distinct positive integers,

Positivity conjecture. For $d\geq 2$, the trigonometric polynomials $\mathcal T_n^{(d)}(t)$ are positive for every $t\in\mathbb R$. Positivity of these polynomials is suggested as…

Let $(S^1)^n$ be the $n$-torus, let $\mathcal F_r$ be the set of all closed geodesic balls of radius $r$ in $(S^1)^n$, and let $\mathcal P((S^1)^n,\mathcal F_r)$ denote the associa…

Let $n$ be a positive integer and let $\alpha_1,\ldots,\alpha_{2n}$ be real numbers satisfying … Consider the alternating exponential sum $\sum_{j=1}^{2n}(-1)^j e^{i\alpha_jt}$. Al…

The paper considers extremal nonnegative trigonometric polynomials for the Rogosinski–Szegő estimate of the second coefficient, distinguishing representations involving odd and eve…

Uniqueness conjecture. $P_n(\theta)$ is the only trigonometric polynomial of order $n$ satisfying (5.59). The claim proposes uniqueness within that class, beyond the explicitly ver…

Special-value conjecture. The following equivalences hold:

Goh–Wildberger conjecture. There are polynomials $\Phi_d(x)\in\mathbb{Z}[x]$, $d\geq 1$, such that, for every $n\geq 1$,

Let $F:(\mathbb{R}/\mathbb{Z})^M\rightarrow\mathbb{C}$ be a trigonometric polynomial, let $\mathfrak S$ be its finite frequency support, and for $\boldsymbol b\in\mathbb{Z}^M$ defi…

Let … where the coefficients $a_j$ are real and $a_1=1$. Odd conjugate-polynomial conjecture. One has … and the solution is unique, namely … This is an extremal problem for conjuga…

Let $h$ denote the maximum gap between the scattered data points in $T\subseteq[-1,1]$, and let $M$ be the truncation parameter in the weighted $\ell^1$ minimization method describ…

Let … where $a_k,b_k$ are independent Gaussian standard random coefficients, and let $Z_n(u)=\mathfrak{h}^0(N_u)$ denote the number of zeroes of $u$. Write $E_n$ for the expectatio…

Let $F_N(x)=\sum_{k=1}^N a_k e^{i n_k x}$ be a trigonometric polynomial with $0<n_1<n_2<\cdots<n_N$, $N\geq 2$, and $|a_k|\geq 1$ for $1\leq k\leq N$. Littlewood's conjecture. Ther…

Let $N\geq 1$, let $D_N(z)=1+z+\dots+z^{N-1}$, and let $f(z)=c_0+c_1z^{n_1}+\dots+c_{N-1}z^{n_{N-1}}$ be an $N$-term trigonometric polynomial with $0<n_1<\dots<n_{N-1}$ and…

Natural-limit conjecture for p-concentration. Based on heuristic arguments and calculations, the authors conjectured that $1$ should be a natural limit for the value of $p$ in $p$-…

Anderson–Ash–Jones–Rider–Saffari conjecture. For $p=1$, no positive constant $\gamma_1$ with this property exists.

The comparison conjecture. There is an absolute constant $c$ such that, for every $n$,

Let $q$ and $m$ be positive integers with $m>q^{2}$, let … and let $f$ be an idempotent trigonometric polynomial. The $L^1$ concentration-failure conjecture. There is an absolute c…