$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'm reading this article by Boyd and Chua [1], in which they prove the approximability of arbitrary time-invariant (TI) operators (or filters) with polynomials of some basis filters with separation and *fading memory* (FM) properties. This shows that those operators, indeed have fintie Volterra expansion approximates.

Consider the following:

- $C(\Bbb R)$ as the space of bounded continuous functions $u: \Bbb R \to \Bbb R$,
- $\norm u = \sup_{t \in \Bbb R} |u(t)|$ being their norm,
- $\Bbb R_-$ as $\{t : t \le 0 \}$, with analogous definitions for $C(\Bbb R_-)$ and $\norm\cdot$ as above,
- and time-invariant functionals $F: C(\Bbb R_-) \to \Bbb R$ and time-invariant operators $N: C(\Bbb R) \to C(\Bbb R)$.

- $K=\bigl\{u\in C(\Bbb R): \abs{u(t)}\le M_1\land\abs{u(s)-u(t)}\le M_2(s-t)\;\forall t, s\in\Bbb R, t<s\bigr\}$: explicitly, $K$ is the space of bounded uniformly Lipschitz continuous functions. Functions belonging to $K$ are called
*signals*.

- $K_-=\{u\in K : u(t)=0\text{ if }t>0\}$: Boyd & Chua prefer to define $K_-$ by using a "projection" operator $P$ such that $$\DeclareMathOperator{\dmu}{d\!} Pu(t) = \begin{cases} u(t) & t\le 0\\ 0 & t>0 \end{cases} $$ and then noting that $K_- =P K$. Be it noted that $K_-$ is compact in $C(\Bbb R)$ but only with respect to the weighted $\sup$ norm defined as $$ \norm u_w=\sup_{t\le 1} \abs{u(t)w(-t)} $$ where $w:\Bbb R_+\to(0,1]$ is a weight function such that $\lim_{t\to\infty} w(t)=0$ (see below).
- For every TI operator $N$, an associated $F$ is defined by $Fu=Nu_e(0)$, in which $u_e$ is some extension of $u$ from $C(\Bbb R_-)$ to $C(\Bbb R)$.
- Furthermore, the fading memory (FM) property is defined for TI operators on $N \in K \subset C(\Bbb R)$ as the existence of some weight function $w: \Bbb R_+ \to (0,1]$ with $\lim_{t \to \infty} w(t) = 0$ that makes $K_-$ closed w.r.t. the weighted norm $w$. i.e. for any $v,u \in K$, one can find a $\delta$ for any $\epsilon$ such that

$$\norm{u-v}_w:= \sup_{t\le 0} \abs{u(t)-v(t)}w(-t) < \delta \implies \abs{Nu(0) - Nv(0)} < \epsilon$$

*Lemma 2* states that there are some functionals on $K_-$ that separate points. To prove this, a class of functionals $G \in \mathbf G \subset K_-$ is defined by
$$\mathbf{G}=\left\{G(u)=\int\limits_0^{+\infty}g(\tau)u(-\tau)\dmu\tau :u\in K_-\land\int\limits_0^{+\infty}\abs{g(\tau)}w(\tau)^{-1}\dmu\tau\right\}$$
which are (shown to be) continuous w.r.t. the weighted norm $w$ (thanks to the condition introduced above). The authors further construct functions $g_0$ defined by:

$$g_0(t):= [u(-t)-v(-t)] w(t)\exp (-t)$$

and their associated $G_0$ **assuming** that they belong to $\mathbf G$. Then, they show that $g_0$ indeed separates points on $K_-$. However, **I don't understand why $G_0 \in \mathbf G$ in the first place.** $g_0$, by construction, dependends on $u$ and $v$. Thus, it does not look to me that $G_0$ is even time-invariant (simply shifting $u$ and $v$ in time will yield a different $g_0$). So my question is why is the proof of this lemma correct?

**Reference**

[1] Stephen Boyd, Leon O. Chua, "Fading memory and the problem of approximating nonlinear operators with Volterra series" (English), IEEE Transactions on Circuits and Systems 32, 1150-1161 (1985), MR0809696, Zbl 0587.93028, doi:10.1109/TCS.1985.1085649.

`\land`

for conjunction rather than`\wedge`

. While fixing some typos, I edited accordingly. $\endgroup$5more comments