Random sampling and approximation of signals with bounded derivatives

Approximation of analog signals from noisy samples is a fundamental, but nevertheless difficult problem. This paper addresses the problem of approximating functions in \(H_{\gamma , \varOmega }\) from randomly chosen samples, where $$ H_{\gamma , \varOmega }= \bigl\{ f \mid f\mbox{ is continuous on } \overline{\varOmega }, \mbox{and } \|D f\|_{L_{\infty }(\varOmega )} \le \gamma \|f\|_{L_{\infty }(\varOmega ) } \bigr\} . $$ We are concerned with the probability that functions in \(H_{\gamma , \varOmega }\) can be approximated from the noisy samples stably and how they can be approximated.