# Is Bernstein ever used in DSP?

In the early 1900s, Bernstein proved you could approximate any continuous function with the sums of polynomials. I know this is a bit weak for DSP because it isn't strictly periodic, but is Bernstein ever used (or was ever used much in the past) for signal reconstruction or up sampling or anything else DSP related?

• Never heard of Bernstein as a separate entitiy but one class of polynomial approximations / interpolation of continuous functions on finite domains was performed via Lagrange polynomial decomposition during 1900s... – Fat32 May 27 '15 at 8:12
• I once used a result of Bernstein's (not the one cited by you) in a paper titled "The frequency spectrum of pulse width modulated signals" (Signal Processing, vol. 83, Oct. 2003) to show that it is necessary to sample a low-pass signal $x(t)$ with highest frequency $f_s$ at a rate greater than $\pi f_s$ (which is larger than the Nyquist rate of $2f_s$) in order to guarantee that $x(t)$ can be reconstructed perfectly by simply lowpass filtering the natural sampling PWM signal. The result used was that $$\max \left|\frac{dx(t)}{dt}\right|\leq 2\pi f_s\max|x(t)|.$$ – Dilip Sarwate May 27 '15 at 13:12