6. For all functions f,ge L (IR), prove that 7. = . Where f. denote the Fourier transforms of f and g. Define basic wavelet and Integral wavelet transform of a function fe L (IR). Let o, We L (IR) are the basic wavelets. Then prove that: (i) [W(af +Bg)](b,a) = a[W f](b,a) + B[Wg](b.a)vf.ge L (IR) and Va, e IR. (ii) (Wao+Byf)(b,a) = a(W+f)(b,a) + B (Wof)(b,a) v fe L (IR) and V , B e C. (iii) [W] (b+a) = (Wf) () v fe L (IR) and c> 0. 8. Let e L (IR) be a basic wavelet and be a bounded and integrable function. Then show that * be a basic wavelet. 9. Let & be the scaling function of the MRA (V), ez Then satisfies the following conditions: (i) (w) = (ii) (w)=mo ()(), where mo(w) = Ck e-kw is a 2- periodic function. KEZ Imo (w)+ |mo(w+) = 1 a. e. 2