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Nov 5 at 17:17 comment added Liding Yao I don't think your question is that valid since for bad enough tempered distribution $f$ the pairing $(f,\psi_{jk})$ is not defined. I used this Triebel's book as well but I believe there are other litreature that talks about noncompact supported Schwartz wavelet with all moment vanishings.
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S Sep 24 at 5:00 history bounty started PhoemueX
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Sep 13 at 15:55 history asked PhoemueX CC BY-SA 4.0