Discrete representations of standard distributions (Q1911595)

Classical results concerning the distribution (algebraic operations, derivatives, Fourier transform) are considered in the context of non-standard analysis, essentially based on the non-standard derivatives \(D_if(x)= {f(x+\varepsilon e_i)-f(x)\over\varepsilon}\), where \(\varepsilon= {1\over\omega}\) with \(\omega\) an unbounded integer and \((e_1,e_2,\dots,e_n)\) a canonical basis of \(\mathbb{R}^n\). For example, if \(L=\{\sum^n_{i=1} x_i\varepsilon e_i; x_i\in\mathbb{Z}\}\), \(X=\{x\in L, x=\sum^n_{i=1} x_i\varepsilon e_i; -{\omega\over 2}\leq\varepsilon x_i<{\omega\over 2}\) \(\forall i=1,\dots,n\}\) and \(y\in X\), one considers the discrete representation of \(\delta\) as given by \(\delta_y(x)=\begin{cases}\varepsilon^{-n} &\text{if }x=y\\ 0 &\text{if }x\neq y\end{cases}\) for which \(({\mathcal F}\delta_y)(x)= e^{-2\pi ixy}\) with the Fourier transform \(({\mathcal F}f)(x)=\sum_{y\in X}e^{-2\pi ixy}f(y)\).