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The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view the Fourier transform as a projection of a vector from the Hilbert space of square-integrable functions on a certain orthonormal basis.

It seems, though, that the Dirac delta-function can be naturally introduced in non-standard calculus. It is a hyperreal function that is $1/dx$ in the interval $[-dx/2, dx/2]$ and zero everywhere else. Here, $dx$ is an infinitesimal hyperreal. But i could not find any treatments of the Fourier transform through non-standard calculus (though it seems to be fruitful and straightforward).

So, my question is: are there any works that treat the Fourier transform through non-standard calculus? Or, if not, are there any intrinsic technical difficulties? And, finally, if it can be done, is it possible to formalize the Fourier transform as a projection of a vector in a linear space of hyperreal functions onto an orthonormal basis?

Bas1l
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    As a side note, in order to define the Dirac delta, you don't need tempered distributions at all. – TZakrevskiy Jan 12 '15 at 09:39
  • I am curious about this as well. If I had to guess, if this is possible then most likely the Fourier transform is the "standard shadow" of a Fourier series for L^2[-T/2, T/2] for T a nonstandard hyperfinite ("infinite") number. (Compare the constructions mentioned here https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf (beginning of chapter 2) and here https://math.stackexchange.com/questions/3375359/why-is-the-infinite-period-fourier-series-the-fourier-transform – Chill2Macht Jan 05 '25 at 18:33
  • One important subtlety is that $L^2(\mathbb{R})$ necessarily corresponds to a "standard shadow" of a proper subset of $L^2[-T/2,T/2]$ for $T$ nonstandard + hyperfinite. Consider the case of constant functions, e.g. $f \equiv 1$, for example. Or even the "standard shadows" of the $e^{2 \pi i n / T}$ basis functions themselves. What would be interesting is if a proper subspace like the Schwatz space for $L^2[-T/2, T/2]$ (https://en.wikipedia.org/wiki/Schwartz_space) is what is needed to get the "standard shadows" to be $L^2(\mathbb{R})$ (or at least a reasonably large subspace thereof). – Chill2Macht Jan 05 '25 at 18:38

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