Mathematical Techniques to Reduce the Width of a Gaussian Peak In the chemical analysis by instruments, the signals of several molecules are overlapped which makes it difficult to determine the true area of each peak, such as those shown in red. I simulated this as a sum of six Gaussians (with some tailing elements)


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*One of the simplest technique is to raise the discrete signal values to any positive power (n>0). The standard deviation of the Gaussians becomes smaller and smaller (C, in blue). The big drawback is that we lose all the original peak area information. The transformed data is highly resolved now at the cost of losing true area information.

*Alternatively, we can add a first derivative of the signal and subtract the second derivative from the original signal i.e., 
Sharpened signal= Original signal +K (first derivative) - J(second derivative)
K and J are small positive real numbers. This neat "trick" maintains the true area because area under the derivatives is negligible (zero in ideal cases).
Do mathematicians use any other transformations which can make each overlapping peak very narrow, yet maintain the original peak areas. I am not interested in curve fitting techniques at this moment. Any pointers to some similar "peak sharpening" transformations would be appreciated which can resolve overlapping signals. 
Thanks.
 
 A: Here is an overview of peak sharpening (resolution enhancement) techniques:
1. even derivative sharpening: $R=Y-k_2 Y''+k_4 Y''''$: works well for symmetric line shapes, reduces the width of a Gaussian peak by 20% and of a Lorentzian peak by 60%
2. first derivative symmetrization: $R=Y+k_1 Y'$: converts an exponentially broadened peak into a symmetric Gaussian, and then the width can be reduced via method 1.
3. power law method: raise each data point to a power $n>1$: moves the peaks apart, increasing the resolution, to be followed by sharpening method 1.
4. deconvolution: effective if the original shape of the peaks has been broadened and/or made asymmetrical by a broadening process that is known or can be measured separately.
A: There are numerous methods in the context of "inverse problems". Assuming that the Gaussian peaks are generated by a convolution of a "spike train" (i.e. a sum of positive Diracs peaks) with a Gaussian (plus some noise), you can do any of the numerous methods for regularized deconvolution. 
Let me just mention variational regularization: Let $g$ be the Gaussian and $\mu$ be the spike train. Then your signal is $f=g*\mu + $ noise. A regularized deconvolution can be obtained by minimizing
$$\|g*\mu - f\| +\alpha\|\mu\|$$
but the choice of norms and the regularization parameter $\alpha>0$ really matter. A simple choice is the $L^2$ norm for the residual and the variation norm (aka Radon norm) for the measure $\mu$. This provably leads to a spiky reconstruction and the process is stable. 
