In the book I am reading, the author explains how we can use the dilation equation to obtain the scaling function through the process of iteration. In particular, we use the dilation equation, which is

$$ \varphi_{n+1}(t) = \sum_{k} h_k \sqrt{2} \varphi_{n}(2t-k)$$

and then, the author says that we "guess" what the scaling function is, i.e., we take a constant function on some interval, and then, by recursion, we produce $\varphi_2(t)$ and then $\varphi_3(t)$ and so on. As we go on, the book says that the function $\varphi_n(t)$ will converge to the true scaling function, the one which corresponds to these coefficients $h_k$. You can also see this here on the 52nd page.

I don't understand why this is so. I mean, I accept that the dilation equation is satisfied for the unique choice of coefficients $h_k$ and the scaling function $\varphi(t)$, but I don't understand why would an arbitrary function converge to this scaling function.

Could anyone elaborate on this? I would be very grateful.


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