Finding family of functions satisfying equation

Question

Is there a family of functions $g_{n}(t)$ (labelled by an integer index $n$ ) that satisfy the following equation?: \begin{eqnarray} \int_{0}^{T}dt g_{n}(t)e^{i(n-m)2\pi t/T}=0, \end{eqnarray} where $n,m\in\mathbb{Z}$, and $g_{n}(t+T)=g_{n}(t)$, e.g. it is a periodic function. The $g_{n}(t)$ family does not need to be necessarily real, they can be complex-valued functions. In fact, it is easier to define $y_{n}(t)=g_{n}(t)e^{in2\pi t/T}$, and so look for orthogonal functions to the complex exponential satisfying: \begin{eqnarray} \int_{0}^{T}dty_{n}(t)e^{-im2\pi t/T}=0. \end{eqnarray} The equation must work for any $n,m\in\mathbb{Z}$ with the $y_{n}(t)\neq 0$.