(Integral of scaling function) Let is a scaling function of an MRA. Then

By the definition ( Multiresolution analysis )-1,2 we have We calculate the last term using the proposition ( Basic properties of Fourier transform )-3: We use the formula ( Property of scale and transport 6 ): Using the propositions ( Dominated convergence theorem ) and ( Fubini theorem ), The collection is an orthonormal basis in . We use the proposition ( Parseval equality ). We use the proposition ( Basic properties of Fourier transform )-4,5. Thus

(Integral of wavelet) Let is a scaling function of an MRA and is the associated wavelet, see the proposition ( Scaling equation 2 ). We have

According to the proposition ( Scaling equation 2 ), Thus According to the proposition ( Scaling equation 3 ), Thus where According to the proposition ( Scaling equation ), an according to the proposition ( Integral of scaling function ) thus

(Properties of sequences h and g) We assume the setup ( Discrete wavelet transform ), and is real-valued. Then

(a) ,

(b) ,

(c) ,

(d) ,

(e) ,

(f) .

of (a). Within the proof the proposition ( Integral of wavelet ) we saw

of (b). According the proposition ( Integral of wavelet ), Change of variable, and then . We use the proposition ( Integral of scaling function ).

of (c),(d),(e). We use orthonormality of for every : We use the formula ( Property of scale and transport 7 ). The property (d) follows similarly from orthonormality of for every , see the proposition ( Existence of orthonormal wavelet bases 2 ). The property (e) follows by considering , see the proposition ( Existence of orthonormal wavelet bases 1 )-c.

of (f). We combine the proposition ( Recursive relationships for wavelet transform )-c with ( Recursive relationships for wavelet transform )-a,b: We now express and in terms of using ( Recursive relationships for wavelet transform )-a,b. Thus