Algebraic Numbers and Harmonic Analysis by Yves Meyer

By Yves Meyer

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The constants, not always the same, de noted by a, b, so, ... depend on A but not on E > 0. , w, a basis of the Z-module generated byA and h : R -t Tnthe continuous homomorphism defined by h(t) = (exp 2niw1t, . , exp 2nio,t) for t E R. , mn = 0); + 1Sj4n (e) for all rational integers m, , . - m,A, = 0 or lm,A, mnAnl2 d ( sup Imjl)-"+l. 2. We shall use bold letters x to denote elements of Rnand put x' = p,(x), = p2(x); Ix"l is the Euclidean norm of x" in Rn-l. To get Theorem V, we use the following lemma; the notation is that of Theorem IV.

Assume that Rn x H2 is produced by G. Then Rn is produced by G. Proof: Define D, to be the set of all d, in G x Rnsuch that for an h in Hz and a din D, d, = (x, y), while d = (x, y, h) where x E G, y E Rnand h E H, . To get the reduced model, let O, be Q n HI and A, be the model delined by H , and Q,. Then A c A, F, where F is finite. Let Q, be the relatively compact subset of Rn which is the projection on Rn of 4, c Rn x Hz and A, the model defined by Rn and Q,. Then A, c A, and A c A, F. Theorem IV is thus proved.

Let F, = B(a) n ( A -A). Starting from 0 it may be checked, by an obviousinduction thatAis contained in the Z-module G, generated by F,. Since F , is finite, the structure theorem of modules on principal rings shows that G, is a direct sum Z o , + ... + Zo,. We now describe a mapping g, : Rn + Rm to which Proposition 14 will be applied: the linear part I of p, will be used to construct the homomorphism h : G, + Rm. ,pm) if x = p , ~ , ... + ~ m w r n ( ~ j1~

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