Let $I = I_{j,k}$ be a dyadic interval. The Haar function associated with I is defined by$$h_{j,k}(x) =\frac{1}{|I|} (\chi_{I_{R}(x)} - \chi_{I_{L}(x)} ),$$ where, $I_{R} = [(2k + 1)2^{-j-1}, (k+1)2^{-j}[$, $I_{L} = [k2^{-j}, (2k+1)2^{-j-1}[.$
Also I want to Show that $$h_{j,k}(x) = 2^{j/2} h_{0,0}(2^{j}x - k).$$
But I feel that this formula contains something wrong in $2^{j/2}$, because I tried the definition given in the question statement for the Haar function (that includes $\chi$)for $j=k=1$, then I tried the definition for the Haar function that we require to prove (that contains $h_{0,0}$) for $j=k=1$, but 1/(the length of the interval was different), (1/the length of the interval as I calculated was 4 but the new formula gives $\sqrt{2}$ )? do you see my problem? did anyone agree with me?
Note that $$h_{0,0}(x)=\begin{cases}-1 & 0≤x<\frac12\\ 1& \frac12≤x<1\end{cases}\qquad |I|\cdot h_{i,j}(x)=\begin{cases}-1 & k≤2^jx<k+\frac12\\ 1& k+\frac12≤2^jx<k+1\end{cases}$$ where I would usually say $|I|=2^{-j}$, but your result needs $|I|=2^{-j/2}$. At any rate compare the two expressions by using the explicit form of $h$ that I wrote out before.