(Recursive relationships for wavelet transform) We assume the setup ( Discrete wavelet transform setup ). For a function we define Then

a. ,

b. ,

c. .

of (a). According to the proposition ( Scaling equation ), We need to extend this relationship to any scale and location. We use the formula ( Property of scale and transport 7 ). Thus Therefore,

of (b). The wavelet is given by the equation thus We use the formula ( Property of scale and transport 7 ). Therefore, We calculate

of (c). According to the definition ( Approximation and detail operators ) and propositions ( Existence of orthonormal wavelet bases 1 ), ( Existence of orthonormal wavelet bases 2 ) We put into and substitute the terms We substitute and . Thus