Oscilloscope design question
When an oscilloscope says it has a 50 MHz bandwidth, that usually means the frequency response is 3dB down (half the power or 71% of the voltage) at 50 MHz.
Although Nyquist says you only need to sample at &gt;100 MHz to reproduce a signal with a 50 MHz bandwidth, that means a signal with no energy above 50 MHz. The usual definition of 3dB down leaves plenty of energy above that.
Sampling at 1 GS/s, we would like no energy above 500 MHz. I'm sure our 50 MHz 'scope could achieve that filtering comfortably, if it wanted to. It probably achieves it accidentally, at least in cheaper 'scopes, by only designing the amplifiers for 50 MHz and letting them roll off at higher frequencies.
By default, most oscilloscopes connect the sampled dots together with straight lines. A 20:1 ratio between the frequency of the displayed waveform and the sample rate is just about enough to get a good-looking sinewave.
The bandwidth limitation on a 'scope probe is a similar beast to that of a scope, it should work up to that frequency, and you don't know what's going to happen above that. Don't assume either a scope or a probe is going to implement a proper filter above its bandwidth specification, don't assume it's going to work to specification above it.