Summary: We consider convergence of thresholding type approximations with regard to general complete minimal systems \(\{e_n\}\) in a quasi-Banach space \(X\). Thresholding approximations are defined as follows. Let \(\{e_n^*\}\subset X^*\) be the conjugate (dual) system to \(\{e_n\}\); then define for \(\varepsilon >0\) and \(x\in X\) the thresholding approximations as \(T_\varepsilon (x) := \sum_{j\in D_\varepsilon (x)} e_j^*(x)e_j\), where \(D_\varepsilon (x):= \{j:|e_j^*(x)|\geq \varepsilon \}\). We study a generalized version of \(T_\varepsilon\) that we call the weak thresholding approximation. We modify the \(T_\varepsilon (x)\) in the following way. For \(\varepsilon >0\), \(t\in (0,1)\), we set \(D_{t,\varepsilon }(x) :=\{j:t\varepsilon \leq|e_j^*(x)|<\varepsilon \}\) and consider the weak thresholding approximations \(T_{\varepsilon ,D}(x) := T_\varepsilon (x) +\sum_{j\in D} e_j^*(x)e_j\), \(D\subseteq D_{t,\varepsilon }(x)\). We say that the weak thresholding approximations converge to \(x\) if \(T_{\varepsilon,D(\varepsilon )}(x) \to x\) as \(\varepsilon \to 0\) for any choice of \(D(\varepsilon )\subseteq D_{t,\varepsilon }(x)\). We prove that the convergence set \(WT\{e_n\}\) does not depend on the parameter \(t\in (0,1)\) and that it is a linear set. We present some applications of general results on convergence of thresholding approximations to \(A\)-convergence of both number series and trigonometric series.