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I'm trying to understand more about wavelets so I went and read the wikipedia article and some other papers on the topic. I have learned that wavelet functions are compact supported and belong to the space $L^1(\mathbb{R})\cap L^2(\mathbb{R})$ i.e. $$\int^\infty_{-\infty}|\psi(t)|\,dt<\infty\quad\rm{and}\quad\int^\infty_{-\infty}|\psi(t)|^2\,dt<\infty.\qquad\qquad(1)$$ However I don't see why the previous conditions imply that $$\int^\infty_{-\infty}\psi(t)\,dt=0\quad\rm{zero\,mean\,condition}\qquad\qquad (2)$$ and $$ \int^\infty_{-\infty}|\psi(t)|^2\,dt=1.\qquad\qquad(3)$$ The previous equations appear in https://en.wikipedia.org/wiki/Wavelet#Mother_wavelet so I would like to know why (1) implies (2) and (3). Thank you for your help.

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    $\begingroup$ The article means to say that conditions (2) and (3) are required of the mother wavelet. (1) does not imply these automatically are always satisfied but does make the conditions (2) and (3) meaningful to require. $\endgroup$ – cauchyproblem Jan 29 '16 at 2:46

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