I have a question about approximations to identity.

Thm A continuous function on $S^1$ with $\hat{f}(n)=0$ for all $n$ is identically zero.

The Kernel they use is $p_k(\theta)= [\epsilon + \cos \theta]^k$. in Stein's textbook, a good kernel has the property

$$\lim_{n\to \infty}(K_n * f)(x) = \int_0^1 K_n(x-y)\, f(y) dy= f(x)$$

Is the function I've described a good kernel? How well can we reconstruct $f(x)$ from it's convolution?

Good Kernel's Properties

  • $\begingroup$ In general, one cannot expect to recover a function from a smoothed version of it... But perhaps you need less for the issue at hand. $\endgroup$ – paul garrett May 15 at 23:19

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