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In the book "An Introduction To Wavelet Analysis" by David F. Walnut, there is,
Exercise 7.45. Show that if $\psi(x)$ is a wavelet, then $\sum\limits_{j}{\left|\hat{\psi}(2^j\gamma)\right|^2} = 1$
Here, $\psi(x)$ is the mother wavelet function, and $\hat{\psi}(\gamma)$ is the Fourier transform of $\psi(x)$. $\hat{\psi}(\gamma) = \int_{-\infty}^{\infty}\psi(x)e^{-2{\pi}i{\gamma}x}dx$
I can not prove it.
But testing with Haar Wavelet by Matlab, it seems right.
Can someone tell me if it's right, and How to prove it?
And thank you for your time!

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  • $\begingroup$ Are you sure that's the right question? By orthonormality of the integer translates of $\psi$, $\sum_j |\hat{\psi}( \gamma - j)|^2 = 1$ a.e. But that formula you have there looks a little weird to me. $\endgroup$ – Michael Aug 22 '13 at 1:18
  • $\begingroup$ Do you need $\psi$ to be an orthonormal basis? $\endgroup$ – freak_warrior Feb 4 '14 at 3:39

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