consider an interesting real analysis question:

define average operator on $[0,1]$:

$A_{\epsilon} f (x) = \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f(y) dy , f \in BV[0,1] $

( may clarify here: $f $ should be regarded as periodic function on $R^1$, i.e. $ f \in BV(S^1) $ )

we know $||A_{\epsilon}f -f||_1 \to 0 $ when $ \epsilon \to 0$.

However its convergence rate is not known in general since the information of $ f $ is not specific, and $ f, \epsilon $ may twist together in convergence rate.

can we find a non trivial convex set $ C \subset BV[0,1] $ s.t. exists $ \gamma>0 $, for all $f \in C $, s.t.

$ ||A_{\epsilon}f -f||_1 \le ||f||_1 \cdot \epsilon^{\gamma} $.

Thanks!

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