$\{c_n\}_{n\in Z}$ is a complex sequence, $S_n(x)$ and $K_m(x)$ are defined as follows $$S_n(x)=\sum_{k=-n}^{n}c_ke^{ikx}$$ $$K_m(x)=\frac{1}{m}\sum_{n=0}^{m-1}S_n(x). $$ $\{K_n\}$ is bounded in $L^p((-π,π))$,$1＜p≤+∞$. Show that there exists $f\in L^p((-π,π))$ s.t.$$ c_n=\frac{1}{2π}\int_{-π}^{π}fe^{-inx}dx, $$ for all $n\in Z$.
I already konw there exist a subsequence of ${K_{n_{l}}}$ weakly converge to some point in $L^p((-π,π))$.But what can I do next? Could someone give me some details? Thank you!