Calculate Haar wavelet expansion by hand

Question

I have been asked to find the wavelet expansion of $$ f(x)= x^2 \text{ for } 0\leq x<1, \quad 0 \text{ otherwise.} $$ I know I have to take the scalar product, and I should use the Haar wavelet. All I have to do is calculate the first couple of coefficients, the problem is I don't know how to start.

Answer 1

The wavelet coefficient will be $$\int_{0}^{1}x^{2}\psi_{j,k}(x)\mathrm{d}x=\int_{0}^{1}x^{2}(\varphi_{j+1,2k}(x)-\varphi_{j+1,2k+1}(x))/\sqrt{2}\;\mathrm{d}x=2^{j/2}\left(\int_{k/2^{j}}^{(2k+1)/2^{j+1}}x^{2}\mathrm{d}x-\int_{(2k+1)/2^{j+1}}^{(k+1)/2^{j}}x^{2}\mathrm{d}x\right),$$ for $0\leq k\leq 2^{j}-1,$ at scales $j\geq 1.$ Now this is just a matter of integrating.