I can't get a problem to work out the way it is supposed to according to my book.
First, a relevant theorem for the problem:
THEOREM
Suppose $\{V_j; j \in Z\}$ is a multiresolution analysis with scaling function $\phi$. Then the following scaling relation holds:
$$\phi(x) = \sum_{k \in Z} p_{k} \phi(2x - k), \quad \text{where } p_k = 2 \int_{-\infty}^{\infty} \phi(x) \overline{\phi(2x - k)} dx$$
Now for the problem:
PROBLEM
Let $\varphi(x)$ be the "tent function," where
$\varphi(x) = x + 1 \text{ for } -1 \leq x \leq 0$
$\varphi(x) = 1 - x \text{ for } 0 < x \leq 1$
$\varphi(x) = 0$ otherwise
Find the scaling relation for $\varphi(x)$ at $p_0$.
ATTEMPT AT SOLUTION
Based on the theorem, I attempted to set this up as follows:
$$p_0 = 2 \int_{-1/2}^{0} (x+1)(2x + 1) dx + 2 \int_{0}^{1/2} (1-x) (1-2x) dx$$
$$ = \frac{5}{12} + \frac{5}{12} = \frac{5}{6}$$
So my final answer would be:
$$\frac{5}{6}\varphi{(2x)}$$
However, according to the answer at the back of the book, the answer should simply be $\varphi(2x)$. So where I get $p_0 = \frac{5}{6}$, I should have gotten $p_0 = 1$. If anyone sees what I'm doing wrong here, I would be extremely grateful!
