Optimal sine and sawtooth inequalities (Q2670651)

The authors study some extremal problems in Fourier analysis of the following type. For each positive integer \(m\), what is the maximal value of \(\sum_{k=1} ^m a_k\) under the constraint that \(\sum_{k=1} ^m a_k \sin(kx) \leq 1\) for every \(x\in\mathbb R\)? The authors solve this problem exactly by identifying, for every \(m\), an explicitly given sequence of nonnegative numbers \(\{a_1,\ldots,a_m\}\) such that \(c_m:=\sum_{k\leq m} a_k \sim \frac{2}{\pi}\log (m+1)\) and this is best possible; a very precise asymptotic for \(c_m\) is given in the paper. The analogous problem is studied and solved for the sawtooth function, namely the case where \(\sin(kx)\) is replaced by \(g(kx)\) with \(g(x):=x+\lfloor \frac12 - x\rfloor\) with \(x\in \mathbb R\). It is again the case that the maximal value of \(c_m \sim \log m\) can be achieved and the authors provide an exact solution. The authors exploit the following linear programming duality. If \(\mu\) is a probability measure on the real line then consider the expectation \[ \mathbb E_{\mu}g(kx):=\int_{\mathbb R}g(kx)\, \mathrm{d}\mu x,\qquad k\in\mathbb N. \] It turns out that the the bound \(\min_{1\leq k \leq m}\mathbb E_{\mu} g(kx)\leq \lambda \) holds for all \(\mu\) (and some \(\lambda\)) if and only if there exist non-negative coefficients \(a_1,\ldots,a_m\) with \(\sum_{1\leq k\leq m}a_k\geq \lambda^{-1}\) and such that \(\sum_{k=1} ^m a_k g(kx)\leq 1\). Roughly speaking, the main theorems in the paper minimize the corresponding expectation \(\mathbb E_{\mu}g(kx)\) by \(c_m^{-1}\). The results in this paper are motivated by problems in algebraic geometry and in particular by the problem of minimizing a notion of volume for smooth complex projective varieties of general type; see for example [\textit{E. Ballico} et al., Commun. Algebra 41, No. 10, 3745--3752 (2013; Zbl 1288.14029)].