Lemma 4.5 — $\|\mathcal S\|<1+\alpha^{-1}$

Statement (IntMul.HvdH.lemma_4_5). Let $s, t$ be natural numbers with $s \neq 0$ and $t \neq 0$, with $s < t$, and with $\gcd(s,t) = 1$. Let $\alpha$ be a real number with $\alpha > 0$. Then the operator norm of the linear map $\mathcal{S} = \mathcal{S}_{s,t,\alpha} : \mathbb{C}^s \to \mathbb{C}^t$ defined below satisfies the strict inequality

The spaces and the norm. $\mathbb{C}^n$ here means the space of functions $\mathbb{Z}/n\mathbb{Z} \to \mathbb{C}$, indices taken as residues $0,1,\dots,n-1$. Each such space carries the supremum norm $\|u\| = \max_{0 \le j < n} |u_j|$ (with $|\cdot|$ the complex modulus). The norm $\|\mathcal{S}\|$ is the operator norm of $\mathcal{S}$ as a map between these sup-normed spaces, i.e. $\|\mathcal{S}\| = \sup\{\|\mathcal{S}u\|_\infty : \|u\|_\infty \le 1\}$ (the $\ell^\infty \to \ell^\infty$ operator norm). For a matrix $A$ this equals the maximum absolute row sum $\max_k \sum_r |A_{k,r}|$.

The map $\mathcal{S}$, unfolded. $\mathcal{S}$ is the linear map given by the $t \times s$ complex matrix $A$, with rows indexed by $k \in \{0,\dots,t-1\}$ and columns by $r \in \{0,\dots,s-1\}$, whose entries are the real numbers (viewed as complex numbers)

so that $(\mathcal{S}u)_k = \sum_{r=0}^{s-1} A_{k,r}\, u_r$. Equivalently, writing $j = r + ms$ and reading $u_j$ as $u_{j \bmod s}$,

The sum over $m \in \mathbb{Z}$ is an unconditional infinite sum; by convention it would be assigned the value $0$ if it failed to converge, but for $\alpha > 0$ the Gaussian terms are summable, so it is the genuine series value. All entries $A_{k,r}$ are strictly positive reals. Consequently the claim is equivalent to

Binders and hypotheses.

• $s, t \in \mathbb{N}$, each assumed nonzero (so $s \ge 1$; together with $s < t$ this forces $t \ge 2$).
• $s < t$ (strict).
• $s$ and $t$ coprime. This includes $s = 1$ with any $t \ge 2$.
• $\alpha \in \mathbb{R}$ with $\alpha > 0$ (strict). There is no upper or lower bound on $\alpha$ beyond positivity; both $\alpha \to 0^+$ and arbitrarily large $\alpha$ are covered.

Edge cases and remarks. The hypotheses are jointly satisfiable (e.g. $s=1$, $t=2$, $\alpha=1$), so the statement is not vacuous. The hypotheses $s<t$ and $\gcd(s,t)=1$ do not enter the definition of $\mathcal{S}$; they only restrict which pairs $(s,t)$ the bound is asserted for. The row $k = 0$ is included, where the row sum is $\alpha^{-1}\sum_{j\in\mathbb{Z}} e^{-\pi j^2/\alpha^2}$. The conclusion is a strict inequality ($<$), not $\le$.