I want ti understand how to calculate the scaling coefficient given scaling function.
In the book I read formula such formula for scaling function is proposed.
$\phi(t) = \sum_{n} h(n) \sqrt2 \phi(2t-n)$
And next, in the example for Haar wavelet system there coefficient $h(0) = \frac{1}{\sqrt 2}$, and $h(1) = \frac{1}{\sqrt 2}$
But I can't understand, how the coefficient has been calculated? Is there exist an explicit formula for scaling coefficient given a scaling function? Or maybe we need to proof it somehow
Typically, one finds the coefficients $h(n)$ satisfying certain properties, then from this obtains $\phi$, but we can do the reverse as well. Consider the Fourier transform of the equation you've written. Then $$\hat{\phi}(\xi)=p\left(\frac{\xi}{2}\right)\hat{\phi}\left(\frac{\xi}{2}\right),$$ where $p(\xi):=\frac{1}{2}\sum_{n}h(n)e^{-2\pi i \xi n}.$ Then we may obtain the coefficients $h(n)$ by considering $$p(\xi)=\left(\frac{\hat{\phi}(\cdot)}{\hat{\phi}(\cdot/2)}\right)(2\xi).$$