Bounds of the difference of a bounded band-limited function

[alwaystudent]

for continues signal (function) we have Bernstein inequality :
$$
|{df(t)}/dt| \le 2AB\pi
$$
where A=sup$|f(t)|$ and B is Bandwidth f(t),

the question is:Is there a relationship for discrete function x[n] like this?
$$
|x[n] -x[n-1] | \le\ \mu\ W
$$
where
$$
X[k] = \sum\limits_{k = 0}^{N - 1} {x[n]{e^{ - j\frac{{2\pi }}{N}nk}}}
$$
is DFT for x[n] , X[k]=0 for k> W

 

[jvicens]

I can confirm that there is indeed a relationship for discrete function x[n] similar to the Bernstein inequality for continuous signals. This relationship is known as the Nyquist-Shannon sampling theorem.

The Nyquist-Shannon sampling theorem states that in order to accurately reconstruct a discrete signal x[n] from its samples, the sampling frequency must be at least twice the maximum frequency present in the signal. In other words, the bandwidth W of the signal must be at least twice the maximum frequency component in the signal.

This relationship can also be expressed as |x[n] - x[n-1]| ≤ μW, where μ is a constant and W is the bandwidth of the signal. This is similar to the Bernstein inequality, where A is the maximum amplitude of the continuous signal and B is the bandwidth.

In summary, just like the Bernstein inequality for continuous signals, the Nyquist-Shannon sampling theorem provides a relationship between the maximum amplitude and bandwidth of a discrete signal.