Function Spaces and Wavelets on Domains (EMS tracts in by Hans Triebel

By Hans Triebel

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First we need the corresponding sequence spaces. 32. Let s 2 R, 0 < p Ä 1, 0 < q Ä 1. 161) such that X s;per kD k jfpq ˇ 2jsq ˇ j;G m j m. 163) j;G;m with the usual modification if p D 1 and/or q D 1, where j m is the characteristic function of a cube with the left corner 2 j L m and of side-length 2 j L (a subcube of T n ). 33. 24. w/ and apq be the corresponding sequence spaces. 149). 160), one has the following basic assertion. 34. Let u 2 N. T n /. 153), Proof. 167) are orthonormal. T n /. 167).

31. Let be an arbitrary domain in Rn with 6 Rn and let u 2 N. 132) with Nj 2 N is called an orthonormal u-wavelet basis in L2 . 4 and an orthonormal basis in L2 . /. 32. 29) but also L. Now we adopt the position that L is chosen sufficiently large as described and fixed. This may justify our omission of L in what follows. 33. Let be an arbitrary domain in Rn with 6D Rn . For any u 2 N there are orthonormal u-wavelet bases in L2 . 31. Proof. Step 1. 131). 128). 1/ 2 . 2/ 2 . / is orthogonal to L2 .

3 we introduce the refined localisation s;rloc spaces Fpq . 40). 42) This equality with in place of U is not true in general for arbitrary domains, but for so-called E-thick domains which include balls and cubes. 28. This may justify the above definition including the restrictions of the parameters. 11. 9. Then kf . U /. 12. The case kf . 16, p. 66. 44) as kf . 45). This local homogeneity is a cornerstone of what follows. It will be used to prove the existence of wavelet bases s;rloc in refined localisation spaces Fpq .

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