Functionals of Harmonics Functions ^†

Let ${\mathcal{C}}_T^s$, with $T>0$, be the set of continuous, T-periodic functions $\mathbf{f}:\mathbb{R}\to {\mathbb{R}}^s$, and let $\Gamma :{\mathcal{C}}_T^s\to \mathbb{R}$ be a real functional on ${\mathcal{C}}_T^s$. If $\Gamma$ is n times Fréchet differentiable on ${\mathcal{C}}_T^s$, then it has an n-th order Taylor expansion around $\mathbf{0}$ (see e.g., [1]). Such a Taylor expansion can be obtained as the n-th order truncation of the series where $\mathbf{n}=(n_1,\dots ,n_s)$ and we have introduced the notation

The kernels $c_{n_1,\dots ,n_s}(t_{11},\dots ,t_{s{n}_s})$ are all real, T-periodic, and symmetric in all their arguments.

In this contribution we will prove the following theorem.

In the special case when $\Gamma$ is invariant under time-shift, i.e., $\Gamma \left[\mathbf{f}\right(t+\tau \left)\right]=\Gamma \left[\mathbf{f}\right(t\left)\right]$ for all $0<\tau <T$, we recover the results in [2] where ${\mathcal{D}}_{+}$ denote the set of nonzero solutions of the Diophantine equation $\mathbf{q}·\mathbf{x}=q_1{x}_1+\cdots +q_s{x}_s=0$, whose leftmost nonzero component is positive.