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

$$\begin{array}{cc}\hfill \Gamma \left[\mathbf{f}\right]=& \sum _{n_1=0}^{\infty }\cdots \sum _{n_s=0}^{\infty }\langle c_{\mathbf{n}}(t_{11},\dots ,t_{1n_1},\dots ,t_{s1},\dots ,t_{s{n}_s})\hfill \\ \hfill & \times f_1\left(t_{11}\right)\cdots f_1\left(t_{1n_1}\right)\cdots f_s\left(t_{s1}\right)\cdots f_s\left(t_{s{n}_s}\right)\rangle ,\hfill \end{array}$$

(1)

where $\mathbf{n}=(n_1,\dots ,n_s)$ and we have introduced the notation

$$\langle \Omega (t_1,\dots ,t_r)\rangle =\frac{1}{T^r}{\int }_0^{T}d{t}_1\cdots {\int }_0^{T}d{t}_r\Omega (t_1,\dots ,t_r).$$

(2)

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.

Theorem 1.

Let Γ be a functional with Taylor series (1), and take

$$\mathbf{f}\left(t\right)=\left({ϵ}_{1}cos(q_1\omega t+{\varphi }_1),\dots ,{ϵ}_{s}cos(q_s\omega t+{\varphi }_s)\right),$$

(3)

where $\mathbf{q}\equiv (q_1,\dots ,q_s)\in {\mathbb{N}}^s$ is such that $gcd(q_1,\dots ,q_s)=1$ and $\omega =2\pi /T$. Then,

$$\begin{array}{cc}\hfill \Gamma \left[\mathbf{f}\right]& =C_{\mathbf{0}}\left(ϵ\right)+\sum _{\mathbf{x}\in S_{+}}{ϵ}_1^{|x_1|}\cdots {ϵ}_s^{|x_s|}{C}_{\mathbf{x}}\left(ϵ\right)cos\left(\mathbf{x}·\mathbf{\varphi }+{\theta }_{\mathbf{x}}\left(ϵ\right)\right),\hfill \end{array}$$

(4)

where, $\mathbf{\varphi }\equiv ({\varphi }_1,\dots ,{\varphi }_s)$, $ϵ\equiv ({ϵ}_1,\dots ,{ϵ}_s)$, and functions $C_{\mathbf{x}}\left(ϵ\right)$ and ${\theta }_{\mathbf{x}}\left(ϵ\right)$ do not depend on ϕ and are even in each ${ϵ}_i$, $i=1,\dots ,s$, for every $\mathbf{x}\in S_{+}$. $\mathbf{x}\in S_{+}$ is the set of vectors $\mathbf{x}$ whose leftmost nonzero component is positive.

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]

$$\begin{array}{cc}\hfill \Gamma \left[\mathbf{f}\right]=& C_{\mathbf{0}}\left(ϵ\right)+\sum _{\mathbf{x}\in {\mathcal{D}}_{+}}{ϵ}_1^{|x_1|}\cdots {ϵ}_s^{|x_s|}{C}_{\mathbf{x}}\left(ϵ\right)cos\left(\mathbf{x}·\mathbf{\varphi }+{\theta }_{\mathbf{x}}\left(ϵ\right)\right),\hfill \end{array}$$

(5)

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.