Wavelets I: Haar Wavelets

Let

and for integers j and k. is known as the mother wavelet.

Fact: the functions form an orthogonal family of functions relative to the inner product

Let

The function is called the scaling function. It is related to the wavelets as follows:

1. Find constants such that

   

   This is called the 2-scale equation.

2. For the function

   

   find constants such that

   

   (Hint: eliminate a lot of by consideration of their supports right at the start.)

3. Carry out the same task as in the preceding problem for

   

   (Hint: You can use the previous problem's result to reduce this problem a little. Then you have to calculate the inner products .)