Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $\varphi:H^{-1}\to\mathbb{R}$. For $x\in H^{-1}$ and a mode $k\in\mathbb{Z}^{n}$, the section of $\varphi$ at $x$ along $k$ is the function $\sigma_{x,k}:\mathbb{R}\to\mathbb{R}$, $\sigma_{x,k}(t)=\varphi(x+te_{k})$, which is defined because $x+te_{k}\in H^{-1}$ by The Wick-Square Problem on the Torus: Standing Notation §units. Differentiability of a real function on $\mathbb{R}$ at a point is that of Single-Variable Calculus on an Interval §derivative, with the interval $I=\mathbb{R}$, every point of which is an interior point of it.

1. (Twice differentiable along the modes)¶ $\varphi$ is twice differentiable along the modes if for every $x\in H^{-1}$ and every $k\in\mathbb{Z}^{n}$ the section $\sigma_{x,k}$ is differentiable at every point of $\mathbb{R}$ and its derivative $\sigma_{x,k}':\mathbb{R}\to\mathbb{R}$ is differentiable at $0$.

2. (Mode derivatives)¶ For such $\varphi$, $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$, the first and second derivatives of $\varphi$ along the mode $k$ at $x$ are the real numbers

$$\partial_{k}\varphi(x)=\sigma_{x,k}'(0),\qquad\partial_{k}^{2}\varphi(x)=(\sigma_{x,k}')'(0).$$