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What are these oscillations?
{\infty}\cos (tx)g(x)\,dx\,\,\underset{\approx}{\longrightarrow}\,\,\,\int_{-L}^{L}\cos(tx)g_{\text{int}}(x)\,dx$$
However, when I calculate an approximation to the Fourier transform, I get some odd oscillations … As I have indicated in the picture above, the oscillations have a "period" of about 15.7. …