Posted by student on December 8, 2007, 2:55 pm
In 1(a) where we prove that {K_delta} is a family of good kernels, I'm having trouble verifying condition (iii), that given any eta>0 the integral of |K_delta| for |x|>=eta goes to 0 as delta goes to 0. It seems like we can give an argument that the integral of |K_delta| for |x|<eta approaches the integral of |K_delta| over R^d as delta approaches 0, but I'm not sure how to argue this formally.
Also, for 1(b) I'm unsure how to use the fact that phi is bounded and supported in a bounded set to verify condition (iii) for approximation to the identity, that for all delta>0 and x in R^d |K_delta(x)|<=A*delta/|x|^(d+1).
For Exercise #17, I think the idea is to let f and x be arbitrary and let epsilon>0 be arbitrary, then try to find a ball B containing x such that c/m(B) * (integral of |f(y)| over B) is >= (f*K_epsilon)(x) for some constant c that does not depend on f, x, or epsilon. However, I'm having trouble implementing this idea.
Any advice? Thanks.
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