I am studying the classical book "Ten Lectures on Wavelets" written by Ingrid Daubechies and I do not understand a specific point. I would appreciate it if someone could help me with detailed explication and new references to my study:
Let $\psi(t)$ be a square-integrable function that satisfies the zero mean condition: $\int_{\mathbb{R}} \psi(t) \, dt=0$. There is an implicit affirmation (please, see the fragment of the Daubechies book) that the zero mean condition to $\psi(t)$ implies that $\hat{\psi}(0)=0$ (here, $\hat{\psi}(t)$ is the Fourier Transform of the $\psi$ function). Why this is true? Can I demonstrate this? Besides, how are these conditions implied in Equation 2.4.1? I am a little confused about these connections.
