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What is the definition of an envelope of a function ? For example if we multiply a sinusoid function of certain frequency ($1/f$ < support of bump) with a bump function we get a function whose envelope looks like bump function. The points where the envelope (bump here) and the given function (the product of sine and bump) meet, their derivatives are equal. Is there any formal definition for such an envelope of any given smooth function (say with compact support) ?

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I'm not sure, so will put as a comment, but I believe the envelope is the center of the osculating circle of the curve at a specific point on the curve. If this is the case, then the envelope at a specific $t$ is $\vec r(t)+\frac{1}{\kappa}N$ where kappa is the curvature at t and N is the unit normal vector to the curve. – fdart17 Feb 15 '11 at 5:39
Also see : – fdart17 Feb 15 '11 at 5:56
up vote 1 down vote accepted

Usually, envelope is connected with a curve families (I imagine you know it), your situation is different..

We can write what you said, that is typical in Signal Theory: modulation of a sinusoidal signal with a generic function. In your situation the envelop is a sort of a continuous extension of the succession of local maximum values for your function.

Mathematically (and here I wish this can help you) we have a swinging signal modulated from a unknown function $f(t)$: how can we obtain this envelope? In your situation, we suppose the Fourier transform of $f(t)$ is different from $0$ in a limited domain $D$, and the frequency of the sinusoidal function $\omega_0> \frac{\max_{\omega}D-\min_{\omega}D}{2}$ $$ f(t)\cos(\omega_0 t + \phi)= f(t)\frac{e^{i(\omega_0 t+\phi)}+e^{-i(\omega_0 t+\phi)}}{2} $$

We know that a product in $x$-variables is a convolution in conjugated variables for Fourier Transform: $$ \frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)\cos(\omega_0 t + \phi) \,dt= \frac{1}{2}\int_{-\infty}^{+\infty}\hat{f}(\omega-\omega')\Bigl[e^{i\phi} \delta(\omega'+\omega_0)+e^{-i\phi}\delta(\omega'-\omega_0)\Bigl]d\omega' $$ $$ =\frac{1}{2}\Bigl[e^{i\phi}\hat f(\omega+\omega_0)+e^{-i\phi}\hat f(\omega-\omega_0)\Bigl] $$ that is proportional to two summed Fourier transform of $f(t)$ translated by $\mp\omega_0$. The Fourier transform of $f(t)$ is different from $0$ in a limited domain $D$, and the frequency of the sinusoidal function $\omega_0$ is higher than $\frac{\max_{\omega}D-\min_{\omega}D}{2}$ for hyp. So the product of this with Heaviside function $\Theta(\omega)=0 $ if $\omega<0$ and $1$ if $\omega>0$ ($\omega=0$ has got zero measure) is

$$ \frac{1}{2}e^{-i\phi}\hat f(\omega-\omega_0) $$ So we obtain Fourier transform translated of $\omega_0$ of enveloping function, that's FT of $\frac{1}{2} e^{-i(\omega_0 t+\phi)} f(t)$. Fourier distributional anti-transform of $\Theta(\omega)$ is $\hat \Theta(t)=\frac{1}{i\ t}$. In $\omega$ domain the product $=\frac{1}{2}\Theta(\omega)\Bigl[e^{i\phi}\hat f(\omega+\omega_0)+e^{-i\phi}\hat f(\omega-\omega_0)\Bigl]$ is the Fourier transform of convolution; then:

$$ f(t) = -2i e^{i(\omega_0 t+\phi)} {\int_{-\infty}^{+\infty}f(\tau)\cos(\omega_0 \tau + \phi)\frac{1}{t-\tau} \,d\tau} $$

that connects enveloping functions to enveloped sinusoid. This is about the physical process that happens in a lock-in, for filter signal from rumor. Remember that this integral has distributional meaning.

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