Professor in class stated this:
Let $G(x)$ be the distribution function of a Laplace(0,1) R.V., and $\Psi(t)$ its characteristic function. For each $\epsilon >0$, Define $G_\epsilon=G(\frac{x}{\epsilon}).$ Set $H_\epsilon=F* G_\epsilon$, where F is a general distribution function then
$\lambda_\epsilon(t)=\int_{-\infty}^{\infty} e^{itx}dH_\epsilon(x)=\varphi(t)\Psi(\epsilon t)$, where $\varphi(t)$ is the characteristic function of F .
I have tried to prove this too little avail, can someone guide me please. This was part of the proof for the inversion formula for characteristic functions.
I would approach this problem with random variables. Suppose $X$ is your Laplace(0,1) rv, suppose $Y$ has distribution function $F$, suppose $Y$ is independent of $X$. Let $Z=Y+\epsilon X$. The characteristic function of $Z$ is $E[e^{itZ}]=E[e^{itY}e^{it\epsilon X}]=E[e^{itY}]E[e^{it\epsilon X}]$ by independence. Finally, tie these steps to the various functions $\Psi$, $\varphi$ and so on in your original statement.
This approach assumes you know the connection between convolution of distribution functions and addition of independent random variables; equivalently, the connection between the characteristic function of the convolution and those of the individual distributions.