Cardinal sine envelop I am looking for an envelope of the function $t\mapsto \mbox{sinc}^2(t)$, or at least an approximation of such an envelope. Is there a known envelope function? Here's an illustration of $\mbox{sinc}^2(t) = \frac{\sin^2(\pi t)}{(\pi t)^2}$ (blue), and of one of its envelopes (red), which I drew by hand. ![sinc2 function and envelope]( More generally, I was wondering, how does one calculate an envelope?

An envelope is defined for a parametric family of functions and not for a single member. It is obtained by eliminating the parameter between the function equation and that obtained by differentiating over the parameter.

For example, consider the family

$$y=\frac{\sin^2\lambda t}{t^2}.$$ The aformentioned derivative is

$$0=\frac2t\sin\lambda t\cos\lambda t,$$ which implies $\sin^2\lambda t=0$ or $1$.

Hence the two envelopes

$$y=\frac1{t^2},\\\y=0.$$