Causal Signal - Fourier Transform or Laplace Transform I am dealing with a physics problem which is related to signal processing. The problem requires me to calculate the instantaneous force acting on a body which depends on some physical parameter $x$. Assume that $x(t)$ is periodic in time for the moment. Since $x(t)$ is periodic, then it can be expanded as a Fourier series with different frequency components (and it doesn't really matter if $x(t)$ is causal). The calculation for the instantaneous force involves adding a complex phase shift (which may depend on the frequency) to each of the frequency component. To do that, I can use the convolution theorem and take the convolution of $x(t)$ with some kernel $\kappa(t)$ whose Fourier transform gives me the required phase shifts, i.e. $\tilde{\kappa}(\omega) \propto e^{i\delta(\omega)}$ where $\delta(\omega)$ is the phase shift. 

Now if in reality $x(t)$ is not periodic and is causal since I only know its values in the past, can I still apply the same kernel to get the instantaneous force? I have been told that I should use Laplace transform instead of Fourier transform. I see the point of it being bilateral by definition, but I am not sure how it is actually different to Fourier transform. Does applying the convolution theorem to a causal signal still give me the desired phase shifts?
What you want is an all-pass filter with frequency response

$$H(\omega)=e^{j\phi(\omega)}\tag{1}$$

where $\phi(\omega)$ is the desired phase shift (and $j$ is how we denote the imaginary unit over here). This system is called an all-pass filter because clearly $|H(\omega)|=1$ holds.

The type of input signal is irrelevant, it can be periodic, non-periodic, causal, or non-causal; if you filter it with a linear time-invariant (LTI) filter with a frequency response given by $(1)$ then the desired phase shift will be achieved.

Your problem is the (causal and stable) realization of such a filter. In general, for a given phase shift $\phi(\omega)$ the frequency response given by $(1)$ cannot be implemented exactly; it can only be approximated.