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Sampling points are $\mathbb{Z}$. Sampling theorem tells us that functions with bandwidth lower than $\frac{1}{2}$ will have no aliases. Take functions with pure frequency $\frac{1}{4}$ as an example.Consider $y=sin(\frac{\pi x}{2})$ and triangle wave $$y=(-1)^k(x-2k),x\in[2k-1,2k+1],k\in\mathbb{Z}$$. Don't they coincide on sampling points? Aren't they aliases? What does sampling theorem say about that situation?

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Your triangle wave is not band limited. It has infinite bandwidth.

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  • $\begingroup$ So there is nothing such as a bandlimited triangle wave? It's really hard to think of sampling theorem in time domain. $\endgroup$ – user33869 Jun 17 '12 at 15:14
  • $\begingroup$ @user33869: it will stop being triangle if you limit bandwidth, I suppose. $\endgroup$ – Violet Giraffe Jun 17 '12 at 16:11
  • $\begingroup$ @user33869: sharp changes means it has arbitrarily large frequency components. The sharpest of all, the Delta function has a no where zero flat frequency spectrum (So its frequency spectrum is the smoothest of all) $\endgroup$ – bleh Jun 26 '12 at 7:37

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