# how to reconstruct a function from its convolution with a good kernel?

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

• 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. – paul garrett May 15 at 23:19