image processing - Determining Type and Bandwidth of a filter

Given a filter, if it is given as an equation such as:

$$f(x,y) = \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \exp\left(-\frac{x^2 + y^2}{\sigma^2}\right)$$

Or in a kernel such as: $$ \left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & -4 & 1 \\ 0 & 1 & 0 \end{array}\right] $$ How much can I find out about the filter. Specifically can I find out if it is a highpass/ lowpass/ bandpass filter (if so how?), and can I find out the bandwidth of the filter?

If it helps, the context is image processing.

Thanks