From Bessel's inequality, we know that for a partial sum of Fourier Series represented by $$[a_0, \mathbf{a}, \mathbf{b}], \mathbf{a}, \mathbf{b} \in \mathbb{R}^N$$ of a periodic function that is square integrable, we can write: $$\frac{a_0^2}{2} + \sum_{n=1}^{N} (a_n^2 + b_n^2) \leq \frac{2}{T}\int_{0}^{T}[f(\tau)]^2 d\tau.$$ Let's say, we have a bounded function $f(t) \in [-L, L], t \in [0, T],$ then we can extend the above inequality to obtain: $$ \frac{a_0^2}{2} + \sum_{n=1}^{N} (a_n^2 + b_n^2)\leq \frac{2}{T} \int_{0}^{T}[f(\tau)]^2 d\tau \leq \frac{2}{T} L^2 T = 2 L^2.$$
Is the above inequality sufficient also for constructing a bounded function with the said limits?
In other words, if I randomly find $a_0, \mathbf{a}, \mathbf{b}$ such that they satisfy the above inequality, will the resulting function $\tilde{f}(t)$ obtained by the Fourier partial sum of these coefficients: $$\tilde{f}(t) = \frac{a_0}{2} + \sum_{n=1}^{N} (a_n cos(\frac{2n\pi t}{T} - n\pi) + b_n sin(\frac{2n\pi t}{T} - n\pi ))$$ be bounded by $[-L, L]$ as well?