SCHWEITZER 1979 Problem 8

Let $K_n(n=1,2,\ldots)$ be periodical continuous functions of period $2 \pi$ , and write $$k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .$$ Prove that the following statements are equivalent:

(i) $\int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)$ for all $f \in \mathcal{L}_1[0,2 \pi]$ .

(ii) $k_n(f;0) \rightarrow f(0)$ for all continuous, $2 \pi$ -periodic functions $f$ .

V. Totik