oo. 3 (Watson's lemma). 8) hold. 9) holds. 9) can be written as I(x) = n-1 ~ a5 L.. 12) is called the remainder term. 18 The Wavelet Transform We can find an estimate to the remainder term [92]. 13) for some number M11 > 0. 12) is bounded by 18n(x)l :(; Mn fooo tn+a-le-(x-a)tdt = Mn r(n+a) = O(x-a-n). 15) The Mellin transform technique We often use Mellin transform in deriving asymptotic expansion of integrals.

3. 1. Let lfl be an r, odd(resp. 2). Let FE w;,wr (JR. 9). 40) to find a boundedness result for the wavelet transform. 2) The above change in order of integration can be justified as follows. We see that I: jt(t)lfl C~b) I =I: dt ~ jt(t)[w(t)] 11Plfl llfllp,w lllflb;allp',w-lfp · C~b) [w(t)t 11PI dt 40 The Wavelet Transform But, for I = (l < p < oo, oo [w(t)] _' P fp ( Jt-bJ)(a-l)p' ( 1 + Jt-bJ)(l-a)p'l (t-b)lp' dt )ljp' 1Jf - I +-a . 8), where E is a constant and Q = s~p{ (I+ lti) 1-a llJI(t)l}.

E and Hf E LP(~n). e. a~Y2 · · · a;;Yn, with Y1 = }'2 = · · · Yn = 2 - p and integrating with respect to a = (a 1, az, ... _)a- 2da }JRn lfl(t)dt+ lrR"+ J'iR"+ r f(b-u) du lrR"+ u a r lfl(-t)dt. v. ( f(b+u)du- ( f(b-u)du) u lJR"+ u f(tb) dt, lJR"+ t- where t- b = (t,- b, )(t2- b2) ... v. 3). 1, we arrive at the following inversion theorem. 1. n), 1I p + 1I p' = < oo. 6). e.. 2. It will be worth while to investigate other results of previous sections to n dimensions and extend to distributions. 1) provided the integral exists, where bE ~n and a> 0.