Say we have a function of time ($S(t)$) of the length $T$, and then a customized impulse response (say $I(t)$) of the length $T+N$. The question is, when $S(t)$ is convolved with $I(t$), what are the possible frequency domain (or alternately time domain) representation of all of the possible resulting functions?
For the case when the convolution is performed circularly, the solution appears immediately clear. If $S(t)$ has zero magnitude components in the frequency domain, they will stay zero no matter how the impulse response is configured. Thus, it's impossible to for example, convert a sinusoidal waveform into any non-sinusoidal waveform using circular convolution.
But the result is a lot less clear for the case when linear convolution is performed. If we limit our horizon to the period of $T$, and completely disregard the tail that the impulse response will generate (for cases where $T>1$, and length of the impulse response $ > 1$ ), what will the answer be then? Are there any limits, or can you transform any function into any other function (provided the tail is thrown away), using linear convolution?
(EDIT: for non-casual impulse responses the "pre-echo" is also thrown away, time shifting is allowed also for such cases).