## 2 Materials and Methods
\]  
of \(\Omega\) is a (total) \(\mu\)-recursive function. [PERSON] introduced the concept of computable real numbers in [1]. Our definition of computable real numbers, and, subsequently, computable complex numbers, is based on computable sequences of rational numbers (p. 14 in [15]). Definition 1: A sequence of rational numbers, \((r_{m})_{m\in\mathbb{N}}\), is called a computable sequence of rational numbers if there exist (total) \(\mu\)-recursive functions \(g,h_{1},h_{2}:\mathbb{N}\to\mathbb{N}\) such that  
\[r_{m}=\frac{(-1)^{g(m)}\cdot h_{1}(m)}{h_{2}(m)+1}\]  
is satisfied for all \(m\in\mathbb{N}\). Analogously, for \(n\in\mathbb{N}\), an \(n\)-fold computable sequence of rational numbers is defined through (total) \(\mu\)-recursive functions \(g,h_{1},h_{2}:\mathbb{N}^{n}\to\mathbb{N}\) in \(n\) arguments. Definition 2: A sequence of complex numbers, \((s_{m})_{m\in\mathbb{N}}\), is called a computable sequence of rational-complex numbers if there exist a pair \(((r_{1,m})_{m\in\mathbb{N}},(r_{2,m})_{m\in\mathbb{N}})\) of computable sequences of rational numbers such that \(s_{m}=r_{1,m}+r_{2,m}\) is satisfied for all \(m\in\mathbb{N}\). Analogously, for \(n\in\mathbb{N}\), an \(n\)-fold computable sequence of rational-complex numbers is defined through a pair of \(n\)-fold computable sequences of rational numbers. Definition 3: A real number, \(x\), is called computable if there exist a computable sequence of rational numbers \((r_{n})_{n\in\mathbb{N}}\) and a (total) \(\mu\)-recursive function \(\xi\colon\mathbb{N}\to\mathbb{N}\) such that \(|x-r_{n}|<2^{-M}\) holds true for all \(n,M\in\mathbb{N}\) that satisfy \(n\geq\xi(M)\). For a triple \((x,(r_{n})_{n\in\mathbb{N}},\xi)\) of this kind, we write \([(r_{n})_{n\in\mathbb{N}},\xi]_{\mathfrak{G}}=x\). Further, we denote the set of computable real numbers by \(\mathbb{R}_{\mu}\). Definition 4: A complex number, \(z\), is called computable if there exist a computable sequence of rational-complex numbers \((s_{n})_{n\in\mathbb{N}}\) and a (total) \(\mu\)-recursive function \(\xi\colon\mathbb{N}\to\mathbb{N}\) such that \(|z-s_{n}|<2^{-M}\) holds true for all \(n,M\in\mathbb{N}\) that satisfy \(n\geq\xi(M)\). For a triple \((z,(s_{n})_{n\in\mathbb{N}},\xi)\) of this kind, we write \([(s_{n})_{n\in\mathbb{N}},\xi]_{\mathfrak{G}}=z\). Further, we denote the set of computable complex numbers by \(\mathbb{C}_{\mu}\). In Definitions 3 and 4, the \(\mu\)-recursive function \(\xi\) provides a computable way to control the approximation error \(|x-r_{n}|\), \(n\in\mathbb{N}\), \(|z-s_{n}|\), \(n\in\mathbb{N}\), respectively. In this case, the convergence of \((r_{n})_{n}\) and \((s_{n})_{n\in\mathbb{N}}\) to \(x\) and \(z\), respectively, is referred to as effective, and the function \(\xi\) is called a corresponding effective modulus of convergence. In the following, we will extend the established concepts of computability to the spaces \(\ell(\star)\) and \(\mathcal{B}(\star)\). To this end, we analogously write  
\[\mathcal{C}\ell(\star):= \begin{cases}\mathcal{C}\ell_{0}^{\infty},&\text{if }\star=\infty,\\ \mathcal{C}\ell^{1},&\text{otherwise},\end{cases} \qquad\mathcal{C}\mathcal{B}( \star):= \begin{cases}\mathcal{C}\mathcal{B}_{0,\pi}^{\infty},&\text{if }\star=\infty,\\ \mathcal{C}\mathcal{B}_{\pi}^{1},&\text{otherwise},\end{cases} \]  
with the relevant formal definitions following below. Further, we employ an enumeration \(\
u:\mathbb{N}\to\mathbb{Z},\ n\mapsto\
u(n)\) of the integers \(\mathbb{Z}\), defined through  
\[\
u(n):=\frac{1}{2}(-1)^{(n\mod 2)}\big{(}n+(n\mod 2)\big{)}.\]  
**Definition 5**. : _A sequence, \(\mathbf{x}\in\ell(\star)\), is called computable in \(\ell(\star)\) if there exist a computable double sequence of rational-complex numbers \((s_{n,m})_{n,m\in\mathbb{N}}\) and a (total) \(\mu\)-recursive function \(\xi\colon\mathbb{N}\to\mathbb{N}\), such that_  
\[\left\|\mathbf{x}-\sum_{n=0}^{\xi(M)}s_{n,M}\cdot\delta_{v(n)}\right\|_{\star}<\frac {1}{2^{M}} \tag{1}\]  
_is satisfied for all \(M\in\mathbb{N}\).