Let the function $f(t)$ be defined as a linear combination of sinusoidal waves, with different amplitudes, frequencies and phases, i.e.
$$f(t) = \sum_{i=1}^n a_i \sin \left( \omega_i t + \varphi_i \right)$$
with $a_i >= 0$ and $\omega_i >= 0$. I want to find an upper bound for $f(t)$, as a function of $a_i$, $\omega_i$ and $\varphi_i$. Obviously, my starting point is $\sum a_i$ but I wonder if it can be improved.
EDIT: In the event that, in the general case, no better upper bound than $\sum a_i$ can be obtained, I wonder whether such a bound exists for the following situations (independent and/or combined).
- The case where all phases are zero: $\varphi_i = 0 \quad \forall i \in \left[1, n \right]$
- The case where the frequencies are strictly increasing: $\omega_j > \omega_i \iff j > i$, and the amplitudes are strictly decreasing: $ a_j < a_i \iff j > i$
- The case where the frequencies $\omega_i$ are all rational numbers.