Download PDF by P. Wojtaszczyk: A Mathematical Introduction to Wavelets

By P. Wojtaszczyk

Beginning with a close and selfcontained dialogue of the overall development of 1 dimensional wavelets from multiresolution research, this booklet provides intimately crucial wavelets: spline wavelets, Meyer's wavelets and wavelets with compact aid. It then strikes to the corresponding multivariable idea and provides actual multivariable examples. this can be a useful booklet for these wishing to benefit concerning the mathematical foundations of wavelets.

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3. 1/5) we use the method of paramultiplication. This method had been used by J. Peetre and H. Triebel around 1976/77r; see [Peet], Ch. 6 and the references given there. M. Bony and Y. Meyer introduced similar techniques in connection with microlocal analysis, non-linear problems, Calderon-Zygmund operators, etc. They also coined the word "paramultiplication". 1/1) and more general pointwise multiplier assertions. 2/2. Here we need a somewhat special but rather sharp assertion which is the basis of the subsequent considerations.

3, p. 138. Let Ko be a C00 function in Rn with n : \y\ < c}, Ko (0) ± 0, (11) for some c > 0, and K(x) = ANK0(x) for some N € N, where A is the Laplacian in R". The local means are given by K(t9g)(x) = f K(y)g(x + ty)dy9 0 < t < 1, g G ^ ( R " ) , (12) with its obvious counterpart Ko(l,g). If 2iV > max(s,<7p) then Lp\\q j (13) as equivalent quasi-norms. The proof in [Triy] shows that c > 0 in (11) may assumed to be small. We insert (9) in (12) and obtain, by the support properties of f(x — 2/c), -2/c)] {x) = YJ"kK{tJ){x-2k\ K ) \LP =(J2\ak\p)kp\\K(tJ)\Lp\\ U^2akf('-2k)\ \ k J (14) k (15) k and, by (13), finally (10) with Aspq = Bspq.

2/14) and that ap = n^ — l j ; w e put h^ = L^. Let the functions q>h,fk,gk have the same meaning as above. We put / ; = gj• = 0 if j < 0. Of course, ZT (/>&)A denotes the Fourier transform of ^ r (/,g), etc. 4 Holder inequalities 49 Proposition Let pup2,q € (0,oo] and ± = ^ + ^ . f/J Le£ 0 < p < oo. Then U=0 1/9

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