Download 2-perfectm-cycle systems can be equationally defined form=3, by Bryant D. E., Lindner C. C. PDF

By Bryant D. E., Lindner C. C.

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U /k Ä c kg. 52) Proof. U /k kgf jAxpq n p s n p Äc kg. U1 /k /f . kg. Rn /k kf . 53) where we used some standard properties of function spaces in unit balls or unit cubes such as extension and multiplication with suitable cut-off functions. 51). 23) and the other properties described there. 54) for some c > 0 and 1 X X lD0 s s Let F11 D B11 . 32). 14. Let be an arbitrary domain in Rn with 6D Rn . Let % D f%lr g be the above resolution of unity. Then ˚ « s;rloc s;rloc . / D f 2 D 0 . / W kf jFpq .

5, pp. 38–42. Although there is a huge literature about periodic wavelets we could not find papers dealing with wavelet bases in the above setting. We followed here essentially [Tri07c]. 1 Function spaces We fix some notation. Let be an arbitrary domain in Rn . Domain means open set. Then Lp . / with 0 < p Ä 1 is the standard quasi-Banach space of all complexvalued Lebesgue measurable functions f in such that ÂZ kf jLp . 1) (with the natural modification if p D 1) is finite. As usual, D. / D C01 .

Rn / W supp f Then ˚ s AQpq . / D f 2 D 0 . 5) Df. x/ ; s s . Rn /k kf jAQpq s where the infimum is taken over all g 2 AQpq . 3) Df. 2. Part (i) is the usual definition of Apq . / by restriction. The spaces s s n Q x Apq . R /. One has an one-to-one correspondence s s . x / and AQpq . /, written in a somewhat sloppy way as between AQpq s s AQpq . /; . Rn / Lloc since j@ j D 0 and Apq 1 R /. The spaces Apq . / with s < 0 will be not of interest for us. Rn /. Rn / with 1 Ä p; q Ä 1 and s > 0. We give an example looking for spaces on domains with j€j > 0, where € D @ .

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