## 2 Materials and Methods
We denote the set of all such sequences by \(\mathcal{C}\ell(\star)\). Further, if we have \(\mathbf{f}=\mathbf{T}_{\star}\mathbf{x}\) for some \(\mathbf{f}\in\mathcal{CB}(\star)\) (c.f. Definition 6), we write \([(s_{n,m})_{n,m\in\mathbb{N}},\xi]_{\mathbf{x}}^{\star}=\mathbf{f}\) for the triple \((\mathbf{f}_{\star}(s_{n,m})_{n,m\in\mathbb{N}},\xi)\)._  
**Definition 6**. : _A function, \(\mathbf{f}\in\mathcal{B}(\star)\), is called computable in \(\mathcal{B}(\star)\) if there exist a computable double sequence of rational-complex numbers \((s^{\prime}_{n,m})_{n,m\in\mathbb{N}}\) and a (total) \(\mu\)-recursive function \(\xi^{\prime}\colon\mathbb{N}\to\mathbb{N}\), such that_  
\[\left\|\mathbf{f}-\sum_{n=0}^{\xi^{\prime}(M)}s^{\prime}_{n,M}\cdot\mathsf{sinc}_{ v(n)}\right\|_{\star}<\frac{1}{2^{M}} \tag{2}\]  
_is satisfied for all \(M\in\mathbb{N}\). We denote the set of all such functions by \(\mathcal{CB}(\star)\). Further, we write \([(s^{\prime}_{n,m})_{n,m\in\mathbb{N}},\xi^{\prime}]_{\mathbf{\delta}}^{\star}=\mathbf{f}\) for the triple \((\mathbf{f},(s^{\prime}_{n,m})_{n,m\in\mathbb{N}},\xi^{\prime})\)._  
Observe that, generally speaking, linear combinations of sinc-functions are _not_ elements of \(\mathcal{B}^{1}_{\pi}\). However, specific linear combinations of sinc-functions with rational-complex coefficients are, in fact, elements of \(\mathcal{B}^{1}_{\pi}\), and the set of these linear combinations is dense in \(\mathcal{B}^{1}_{\pi}\). For details, we refer the reader to Appendix B, p. 6363f in [19] (upon minor adjustments, the proof presented in the reference holds true for the restricted case of rational-complex coefficients). A sequence \(\mathbf{x}\in\mathcal{C}\ell(\star)\) is called an elementary computable if there exists a rational-complex \((2L+1)\)-tuple \((z_{k})_{k\in\mathcal{I}},\mathcal{I}:=\{0,\ldots,2L\}\), such that we have  
\[\mathbf{x}:=\sum_{k=0}^{2L}z_{k}\cdot\delta_{v(k)}.\]  
Analogously, a function \(\mathbf{f}\in\mathcal{CB}(\star)\) is called an elementary computable if there exists an elementary computable sequence \(\mathbf{x}\in\mathcal{C}\ell(\star)\) such that we have \(\mathbf{f}=\mathbf{T}_{\star}\mathbf{x}\). Hence, elementary computable functions are exactly those functions that we can represented by a finite interpolation series with rational-complex coefficients \((z_{k})_{k\in\mathcal{I}}\) in the sense of traditional SSI. For \(\mathbf{x}\in\mathcal{C}\ell(\star)\), a computable double sequence of rational-complex numbers \((s_{n,m})_{n,m\in\mathbb{N}}\), and a (total) \(\mu\)-recursive function \(\xi\colon\mathbb{N}\to\mathbb{N}\), let (1) be satisfied. Then, for all \(M\in\mathbb{N}\), the sequence  
\[\mathbf{x}_{M}:=\sum_{n=0}^{\xi(M)}\underbrace{s_{n,M}\cdot\delta_{v(n)}}_{:=z_{n }}\]  
is an elementary computable with coefficients \(z_{n},\ n\in\mathbb{N}\). The sequence of sequences \((\mathbf{x}_{M})_{M\in\mathbb{N}}\) is called a computable sequence of elementary computable sequences. Further, for all \(M\in\mathbb{N}\), we have \(\left\|\mathbf{x}-\mathbf{x}_{M}\right\|_{\star}<2^{-M}\). In general, a sequence \(\mathbf{x}\in\ell(\star)\) is computable in \(\ell(\star)\) if and only if there exists a computable sequence of elementary computable sequences \((\mathbf{x}_{M})_{M\in\mathbb{N}}\) that converges effectively towards \(\mathbf{x}\), with respect to \(\left\|\cdot\right\|_{\star}\) and a suitable effective modulus of convergence. For \(\mathbf{f}\in\mathcal{CB}(\star)\), a computable double sequence of rational-complex numbers \((s^{\prime}_{n,m})_{n,m\in\mathbb{N}}\), and a (total) \(\mu\)-recursive function \(\xi^{\prime}\colon\mathbb{N}\to\mathbb{N}\), let (2) be satisfied. Analogously, for all \(M\in\mathbb{N}\), the function  
\[\mathbf{f}_{M}:=\Upsilon_{\star}\sum_{n=0}^{\xi^{\prime}(M)}s^{\prime}_{n,M}\cdot \delta_{\
u(n)}=\sum_{n=0}^{\xi^{\prime}(M)}z^{\prime}_{n}\cdot\operatorname{ sinc}_{\
u(n)} \tag{3}\]  
is an elementary computable with coefficients \(z^{\prime}_{n},\ n\in\mathbb{N}\).