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Here $\delta(x)$ is the dirac-delta function.

Are there known transformations $T$ that achieve this for a function $f(x)$ based on a parameter $a$ (say, $0 \le a \le 1$)?

$$T_{a}[f(x)]$$

$$T_{0}[f(x)] = \delta(x)$$ $$T_{1}[f(x)] = f(x)$$

and it should smoothly vary between $a=0$ and $a=1$ (not particular about how it should transition, but simpler the better. Edit: It should not involve a delta function when $a > 0$, assuming f(x) itself doesn't contain any).

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    Good one :) I should have mentioned, but I want to stay away from $\delta(x)$ when $a > 0$. – 2017-01-01
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    I'm not really sure what you're asking for but what about $$f(x)(e^{-x^2})^{1/a-1}$$ – 2017-01-01
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    @juanarroyo, when $a=0$, doesn't this result in $f(x) e^{x^{2}}$ and not $\delta(x)$? – 2017-01-01
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    Use it for $a>0$, and use $\delta (x)$ for $a=0$. – 2017-01-01
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    Sorry I see my notation may have been misleading. It's supposed to be $$f(x)(e^{-x^2})^{\frac{1}{a}-1}$$ – 2017-01-03

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Hint: try a linear combination of $f$ with an approximation to the identity.