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Operations on distributions
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arbitrary Assume bounded carrier compatible continuous function F(x converges almost uniformly converges distributionally converges in square converges strongly converges uniformly converges weakly coordinate Corollary defined definition delta sequence delta-sequence denote Differentiating dimensional distri distribution f(x distributional derivative distributionally convergent dn(x equality equivalent exist an integer exist smoothly exists a.e. Fa(x fact follows formula Fourier transform functions in Rq fundamental sequence gn(x Hence Hilbert transform holds induction inequality inner product interval inside k e Pq Lebesgue integral Lemma limit locally integrable function matrices Moreover nePq obtain open set periodic distributions periodic functions periodic smooth functions pn(x point x0 polynomial of degree positive integer positive number Proof prove rapidly decreasing real number regular operation right hand side satisfy Section sequence Fn(x sequence of distributions sequence of smooth sequence of tempered square integrable functions square mean subset tempered distribution Theorem uniformly convergent write