I am searching for some examples of complete families of functions $\left\{ \phi_m(t) \right\}_{m = 1}^\infty$ on $t \in [0, T]$ that form an absolutely convergent series: $$ \sum_{m = 1}^\infty |\phi_m(t)| < \infty, ~~~~ t \in [0, T]. $$
According to the Weierstrass M-test, $\phi_m$ must satisfy $$ |\phi_m(t)| \leq c_m, ~~ t \in [0, T], $$ such that $$ \sum_{m = 1}^\infty c_m < \infty. $$
Any idea or references would be highly appreciated.
According to the definition of complete system in the link provided, $\{\phi_m\}$ must be an orthonormal system. This implies among other things that$\int_0^T|\phi_m|^2\,dt=1$. Suppose that there is a constant $M>0$ such that $$ |\phi_m(t)|\leq M\quad\forall t\in[0,T]\quad\text{and}\quad \sum_{m=1}^\infty\int_0^T|\phi_m(t)|\,dt\leq M. $$ This holds in particular if $|\phi_m(t)|\leq c_m$ with $\sum_{m=1}^\infty c_m<\infty$. Then $$ \sum_{m=1}^\infty|\phi_m(t)|^2\le M\sum_{m=1}^\infty|\phi_m(t)|,\quad0\le t\le T. $$ Integrating we get $\infty$ on the left hand side, and $M^2$ on the right hand side, a contradiction. This shows that no such families exist.