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Daniele Tampieri
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Boyd & Chua 1985: Is the proof of Lemma 2 correct?

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 $u: \Bbb R \to \Bbb R$,
  • $\|u\| = \sup_{t \in \Bbb R} |u(t)|$ being their norm,
  • $\Bbb R_-$ as $\{t | t \le 0 \}$, with analogous definitions for $C(\Bbb R_-)$ and $\|\cdot\|$ as above,
  • and time-invariant functionals $F: C(\Bbb R_-) \to \Bbb R$ and time-invariant operators $N: C(\Bbb R) \to C(\Bbb R)$.
  • $K=\big\{u\in C(\Bbb R): |u(t)|\le M_1\wedge|u(s)-u(t)|\le M_2(s-t)\;\forall t, s\in\Bbb R, t<s\big\}$: 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 respect to the weighted $\sup$ norm defined as $$ \| u\|_w=\sup_{t\le 1} |u(t)w(-t)| $$ 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 $u_e$ os 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

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

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 $$\mathbf{G}=\left\{G(u)=\int\limits_0^{+\infty}g(\tau)u(-\tau)\dmu\tau :u\in K_-\wedge \int\limits_0^{+\infty}|g(\tau)|w(\tau)^{-1}\dmu\tau\right\}$$ which are (shown to be) continious 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.

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