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I know that the Fourier tranform of a Gaussian function is still a Gaussian function.

I also checked Wikipedia:

a closed-form expression is a mathematical expression that can be evaluated in a finite number of operation

But I can't get what exactly the expression 'Gaussians form a closed set wrt Fourier transform' means. Does anyone care to elaborate?

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The term "closed" is not related to closed-form expressions in this context. A set $X$ is closed under an operation $f$ if for each $x$ in $X$, $f(x)$ is in $X$. That is, applying the function $f$ to all of the elements in the set does not produce any new elements. In this case, $X$ is the set of all Gaussians, and $f$ is the Fourier transform.

We can write this in another way:

The image of $f$ under $X$ is defined as: $$ f(X)=\{f(x)|x\in X\}. $$ We say $X$ is closed under $f$ whenever $$ f(X)\subset X. $$

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