Here wikipedia it is said that the Dirac delta could be thought of as $$ \delta(x) = \left\{ \begin{array}{ll} \infty &, x = 0 \\ 0 &, x \ne 0 \end{array}\right. $$ and here that the Fourier-Transform of $\cos(2\pi k_0 x)$ is $$ \frac{1}{2} \left( \delta(k - k_0) + \delta(k + k_0) \right). $$ From computer programs I know that the spectrum gives the intensity of every cos/sin wave in the waveform, but where is the intensive (i.e. one) of the simple cosine waveform represented when i thought of $\delta(k-k_0)$ as being "stretched till infinity" at the point $k_0$?
The Dirac impulse and Fourier transform
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analysis
fourier-analysis
distribution-theory