## 3 Results
We merely aim at finding find proper digital representatives for the continuous-time signals \(\mathbf{f}_{t}\), \(\widetilde{f}_{t,1},\dots,\widetilde{f}_{t,M}\), \(\mathbf{f}_{\text{de,}t,1},\dots,\mathbf{f}_{\text{de,}t,N}\), \(\mathbf{x}_{t}\), \(\widetilde{f}_{t,1},\dots,\widetilde{f}_{t,M}\), \(\mathbf{f}_{\text{de,}t,1},\dots,\mathbf{f}_{\text{de,}t,N}\), since only considering their discrete-time counterparts is insufficient for the reasons we mentioned above. Nevertheless, \(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\bm  
In a communication system of the form (5) and (6), we consider \(\|f_{t}\|_{1}\), \(\|\widetilde{f}_{t,m}\|_{\infty}\), \(\|f_{\mathsf{de},t,m}\|_{\infty}\), \(\|f_{t}*\widetilde{f}_{t,m}\|_{\infty}\), \(\|\widetilde{f}_{re,t}-f_{\mathsf{de},t,m}\|_{\infty}\), and \(\|\widetilde{f}_{re,t}-(f_{t}*\widetilde{f}_{t,m})\|_{\infty}\) relevant properties for all \(t\in\mathbb{N}\), \(m\in\{1,\ldots,M\}\), \(n\in\{1,\ldots,N\}\). Thus, regarding any sufficient representation of signals \(f\in\mathcal{CB}(\star)\) on digital hardware, we require to be able to recover \(\|f\|_{\star}\). In other words, the mapping \(\|\cdot\|_{\star}:\mathcal{CB}(\star)\to\mathbb{R}_{\mu}\), \(f\mapsto\|f\|_{\star}\) has to be computable in the employed machine-readable language. Theorem 3.2: _The mapping \(\|\cdot\|_{\star}:\bar{\mathfrak{x}}_{\star}\to\mathfrak{R}\), \((f,\Lambda_{\bar{\mathfrak{x}}},[\cdot]_{\mathfrak{R}}^{\star})\mapsto(\|f \|_{\star},\Lambda_{\mathfrak{R}},[\cdot]_{\mathfrak{R}})\) is computable._  
Proof: Observe that for \(f\in\mathcal{CB}(\star)\) elementary computable, the mapping \(f\mapsto\|f\|_{\star}\) is computable. That is, for \(N,M\in\mathbb{N}\) and a rational-complex \((M+1)\)-tuple \(\mathbf{z}:=(z_{m})_{m\in\mathcal{I}}\), \(\mathcal{I}=\{0,\ldots M\}\), there exists a computable mapping \((N,z)\mapsto G_{\star}(N,z)\in\mathbb{Q}\), such that  
\[d\Big{(}\widetilde{\varkappa}_{t},\big{(}\widetilde{\varkappa},\Lambda_{\Psi}, [\cdot]_{\Psi}\big{)}\Big{)}:=\big{\|}\widetilde{\varkappa}_{t}-\widetilde{ \varkappa}\big{\|}_{2},\qquad\widetilde{\varkappa}\in\widetilde{\mathbb{R}}^{2}\]2  
holds true (provided the relevant norm exists). For \(\mathbf{f}=[(s^{\prime}_{n,m})_{n,m\in\mathbb{N}},\bar{\xi}^{\prime}]_{\mathfrak{R }}^{\star}\) arbitrary, let the computable sequence \((f_{M})_{M\in\mathbb{N}}\) of elementary computable sequences satisfy (3). Further, for \(M\in\mathbb{N}\), define  
\[d\Big{(}\widetilde{\varkappa}_{t},\big{(}\widetilde{\varkappa},\Lambda_{\Psi}, [\cdot]_{\Psi}\big{)}\Big{)}:=\big{\|}\widetilde{\varkappa}_{t}-\widetilde{ \varkappa}\big{\|}_{2},\qquad\widetilde{\varkappa}\in\widetilde{\mathbb{R}}^{2}\]3  
Then, \((r_{M})_{M\in\mathbb{N}}\) is a computable sequence of rational numbers. Employing the triangle inequality, we obtain  
\[d\Big{(}\widetilde{\varkappa}_{t},\big{(}\widetilde{\varkappa},\Lambda_{\Psi}, [\cdot]_{\Psi}\big{)}\Big{)}:=\big{\|}\widetilde{\varkappa}_{t}-\widetilde{ \varkappa}\big{\|}_{2},\qquad\widetilde{\varkappa}\in\widetilde{\mathbb{R}}^{2}\]4  
Thus, defining \(\xi:\mathbb{N}\to\mathbb{N},M\mapsto M+1\), we have \([(r_{M})_{M\in\mathbb{N}},\xi]_{\mathfrak{R}}=\|f\|_{\star}\). Further, it follows from the SMN-Theorem (c.f. Section 2) that the mapping \(((s^{\prime}_{n,m})_{n,m\in\mathbb{N}},\bar{\xi}^{\prime})\mapsto((r_{M})_{M \in\mathbb{N}},\bar{\xi})\) (with \(((s^{\prime}_{n,m})_{n,m\in\mathbb{N}},\bar{\xi}^{\prime})\) and \(((r_{M})_{M\in\mathbb{N}},\bar{\xi})\) as above) is computable, which concludes the proof. Theorem 3.3: _The mapping \(\|\cdot\|_{\star}:\bar{\mathfrak{x}}_{\star}\to\mathfrak{R}\), \((f,\Lambda_{\bar{\mathfrak{x}}},[\cdot]_{\mathfrak{R}}^{\star})\mapsto(\|f\|_{ \star},\Lambda_{\mathfrak{R}},[\cdot]_{\mathfrak{R}})\) is not computable._  
Proof: The statement follows by contradiction from Lemmas 1 and 2, respectively.