We all know that the space of smooth functions on Euclidean space with compact support is dense in the Lp spaces, for p strictly less than infinity. Now my question is: suppose there is a function f which is in Lp space as well as essentially bounded. Is it possible to find a smooth function g with compact support such that it approximates the original function in Lp norm as well as its supremum controlled by the supremum of f, on the support of g?
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2$\begingroup$ Pretty much whatever you want in this issue, except density with respect to $L^\infty $, is true. $\endgroup$– IanSep 4, 2015 at 12:53
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$\begingroup$ Thanks for your comment. I know it is generally impossible to approximate f in l-infinity. But what I want here is only that the approximating function g has supremum norm less than or equal to that of f. $\endgroup$– Tongou YangSep 4, 2015 at 13:07
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$\begingroup$ Yes, you can do that with no loss. $\endgroup$– IanSep 4, 2015 at 13:55
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1 Answer
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To be more precise than the comments, use the usual approximation by convolution. This will give you a $C^\infty$ function which is bounded by the $L^\infty$ norm of $f$ and is arbitrarily close to $f$ in $L^p$.
Now truncate by a suitable $C_c^\infty$ bump function $\varphi$ which fulfills $0 \leq \varphi \leq 1$.
