On the construction of multivariate (pre)wavelets by de Boor C., DeVore R.A., Ron A.

By de Boor C., DeVore R.A., Ron A.

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Extra info for On the construction of multivariate (pre)wavelets

Example text

Ron Application of this equality, once with v = v, u = vo, and then with the opposite choice, proves that 2-dQo(%B) = B~B~o -- B~oB~ = O. 6) is satisfied and the functions w~, v ~ Vo, are in W. 8, B = lit/, q~ is a trigonometric polynomial, hence so is each B~ and each %. This implies that each % is compactly supported. To show the LE-Stability of the (wv)~Vo, we consider the matrix T := (%,u).... v with diagonal elements zv, ~:= - B~o, v e V, and with off-diagonal elements %, ~o = B~, v e Vo, and with all other entries zero.

152 C. de Boor, R, A. DeVote, and A. , n. Otherwise, Y is trivial. 4) G(y) = 1~ ( 2 , . - iy) -~. m=l In this case, 5~k = ~q~ for every k. For convenience, we define from now on W_~ := ~ fashion the decomposition L:(R) = 5#k, and obtain in this Q -oo ko > -oo), (ok is always a linearly independent generator, and in particular a stable generator of its space.

E~_,B(" + ~(v))B(" + ~(u))~O, ~ ~, = ev_uB('. + o;(v))B(" + or(u)). On the Construction of Multivariate (Pre)Wavelets 159 (Here, we wrote B(. 14) [~*, ~*] = ~, [[~,*,~*~(" + #) ,tt E4r~V = 2 za ~ ev-u(" + #)B(" + a(v) + #)B(" + e(u) + #). 14) corresponding to # and # + ct(v) - a(u) are negatives of one another because of our assumption (b), and the (a(v) - a(u))periodicity of the term B(. + ~(v) + #)B(' + ~(u) + #). Hence Wv, "* wu ^* ] = O, v • u. 15) ^*o ] = 22a [ w^*v , w ~ tz~ 4~zV B('+~(v)+#)z=22a ~ B('+#)2=22aQo(B2).

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