A Banach space \((B, \Vert \cdot \Vert_B )\) has the bounded approximation property if there exists a constant \(C_B \geq 1\) such that for any compact set \(M \subset B\) and \(\epsilon > 0\) there exists a finite rank operator \(F\) on \(B\) with \[ ||| F |||_{B \rightarrow B} \leq C_B \text{ and } \Vert F(f) - f \Vert \leq \epsilon \text{ for all } f \in M. \] Denote by \(\mathcal{S}(\mathbb{R}^d)\) the Schwartz space of rapidly decreasing functions on \(\mathbb{R}^d\). The Banach space \((B,{ \Vert \cdot \Vert_B }) \) is defined to be a minimal tempered standard space (MINTSTA) if the following conditions are satisfied:

1.

The chain of embeddings \[ \mathcal{S}(\mathbb{R}^d) \hookrightarrow (B, \Vert \cdot \Vert_B) \hookrightarrow \mathcal{S}^{'}(\mathbb{R}^d). \] are continuous.

2.

\(\mathcal{S}(\mathbb{R}^d)\) is dense in \((B, \Vert \cdot \Vert_B)\).

3.

\((B, \Vert \cdot \Vert_B)\) is translation invariant, and for some \(s_1 \in \mathbb{N}\) and \(C_1 > 0\) one has \[ \Vert T_x f \Vert_B \leq C_1 \langle x \rangle^{s_1}\, \Vert f \Vert_B, \text{ for all }x \in \mathbb{R}^d, \, f \in B. \]

4.

\((B, \Vert \cdot \Vert_B)\) is modulation invariant, and for some \(s_2 \in \mathbb{N}\) and \(C_2 > 0\) one has \[ \Vert M_y f \Vert_B \leq C_2 \langle y \rangle^{s_2}\, \Vert f \Vert_B, \text{ for all } y \in \mathbb{R}^d, \, f \in B. \]

Here \(\langle x \rangle^{s_1}\) and \(\langle y \rangle^{s_2}\) represent a Beurling weight of polynomial type.
The main result of the paper under review is: Any MINTSTA satisfies the bounded approximation property.