## Appendix A Proofs of Lemmas 1 and 2  
This appendix provides detailed proofs of Lemmas 1 and 2 from Section 2, as well as a brief discussion of the discontinuity of the interpolation operator \(\mathbf{T}_{\star}\). To this end, we establish two additional supplementary results, Lemmas A1 and A2. There exists a computable sequence \((f_{M})_{M\in\mathbb{N}}\) of elementary computable functions in \(\mathcal{CB}^{\infty}_{0,\pi}\) such that \(f_{M}(1/2)\geq 1\), \(\|\mathcal{S}_{\infty}f_{M}\|_{\infty}<\
icefrac{{1}}{{M}}\), and \(\|f_{M}\|_{\infty}\leq C\) are satisfied for all \(M\in\mathbb{N}\) and some number \(C>0\). Observe that Lemma A1 implies the unboundedness (and hence discontinuity) of the interpolation operator \(\mathbf{T}_{\infty}\). For \((f_{M})_{M\in\mathbb{N}}\) as in Lemma A1,  
\[\lim_{M\to\infty}\frac{\|\mathbf{T}_{\infty}\mathcal{S}_{\infty}f_{M}\|_{\infty}} {\|\mathcal{S}_{\infty}f_{M}\|_{\infty}}=\lim_{M\to\infty}\frac{\|f_{M}\|_{ \infty}}{\|\mathcal{S}_{\infty}f_{M}\|_{\infty}}>\lim_{M\to\infty}\frac{1}{1 /M}=\infty \tag{1}\]holds true. Further, it provides an example of a sequence \(\mathbf{x}\in\ell_{0}^{\infty}\setminus\text{dom}(\mathbf{T}_{\infty})\): for all \(N\in\mathbb{N}\), we have  
\[\left\|\sum_{M=1}^{N}\frac{\mathcal{S}_{\infty}f_{M}}{M}\right\| _{\infty}\leq\sum_{M=1}^{N}\frac{1}{M^{2}}=\frac{\pi^{2}}{6},\quad\left\|\sum_{ M=1}^{N}\frac{f_{M}}{M}\right\|_{\infty}\geq\left|\sum_{M=1}^{N}\frac{f_{M}( \
icefrac{{1}}{{2}})}{M}\right|\geq\ln(N), \tag{10}\]  
i.e., there exists \(\mathbf{x}\in\ell_{0}^{\infty}\) with \(\mathbf{x}=\mathcal{S}_{\infty}f_{1}+\
icefrac{{1}}{{2}}\cdot\mathcal{S}_{\infty }f_{2}+\
icefrac{{1}}{{3}}\cdot\mathcal{S}_{\infty}f_{3}+\ldots\), while the sequence \(\mathbf{f}_{1},\;f_{1}+\
icefrac{{1}}{{2}}\cdot f_{2},\;f_{1}+\
icefrac{{1}}{{2} }\cdot f_{2}+\
icefrac{{1}}{{3}}\cdot f_{3}\ldots\) diverges with respect to \(\|\cdot\|_{\infty}\). Proof of Lemma 1.: To begin with, consider a computable sequence of rational numbers \((r_{m})_{m\in\mathbb{N}}\) that satisfies \(r_{m}-2^{-m}<\pi<r_{m}\) for all \(m\in\mathbb{N}\). Further, define  
\[s_{n,M}:=\sum_{k=1}^{\infty}\frac{(-1)^{k}\delta_{k}[\
u(n)]}{ \Delta\big{(}2^{8M}\big{)}},\;\Delta(m):=\sum_{k=1}^{m}\frac{-1}{r_{m}(k-\frac {1}{2})},\;\xi(M):=2^{8M+1}\]  
for all \(n,m,M\in\mathbb{N}\). Then, the pair \(((s_{n,M})_{n,M\in\mathbb{N}},\xi)\) determines a computable sequence \((f_{M})_{M\in\mathbb{N}}\) of elementary computable functions in the sense of Section 2. For \(K=2^{8M}\), \(M\in\mathbb{N}\), we have  
\[f_{M}:=\sum_{n=0}^{\xi(M)}s_{n,M}\cdot\text{sinc}_{\
u(n)}=\frac {1}{\Delta(K)}\sum_{k=1}^{K}(-1)^{k}\,\text{sinc}_{k}\,.\]  
In the following, we will prove that \((f_{M})_{M\in\mathbb{N}}\) satisfies the requirements of the lemma. First, observe that  
\[\text{sinc}_{M}\bigg{(}\frac{1}{2}\bigg{)}=\frac{(-1)^{M}}{\pi( \frac{1}{2}-M)}\]  
holds true for all \(M\in\mathbb{N}\). Consequently, for all \(K=2^{8M}\), \(M\in\mathbb{N}\), we have  
\[f_{M}(\
icefrac{{1}}{{2}})=\frac{\sum_{k=1}^{K}(-1)^{k}\frac{(-1 )^{k}}{\pi(\frac{1}{2}-M)}}{-\frac{1}{r_{M}}\sum_{k=1}^{K}\frac{1}{k-\frac{1}{ 2}}}\geq-\frac{\sum_{k=1}^{K}\frac{1}{(k-\frac{1}{2})}}{\sum_{k=1}^{K}\frac{1 }{\frac{1}{2}-k}}=1.\]  
Next, recall that we have \(r_{N}-2^{-N}<\pi<r_{N}\) and thus \(r_{N}<4\) for all \(N\in\mathbb{N}\). Hence, the inequality  
\[|\Delta(N)|=\frac{1}{r_{N}}\sum_{k=1}^{N}\frac{1}{k-\frac{1}{2}} >\frac{1}{4}\int_{1}^{N+1}\frac{1}{\tau-\frac{1}{2}}\;\mathrm{d}\tau=\frac{1} {4}\bigg{(}\ln\bigg{(}N+\frac{1}{2}\bigg{)}-\ln\bigg{(}\frac{1}{2}\bigg{)} \bigg{)}>\frac{\log_{2}(N)}{4}\]  
is satisfied for all \(N\in\mathbb{N}\).