3

Let $f∈L_1[0,\pi]$. Consider the Fourier coefficient $\{c_n\}$ in trigonometric system $\{\sin(nx)\}$.

1)Is it necessary to converge series $\sum_{n=1}^\infty |с_n|$?

2)Under what conditions does the series is converge $\sum_{n=1}^\infty c_n^2$

As it appears, in 1) I need example for which this series diverges. But I cant come up such example. And I have no idea about 2)

Michael Hardy
  • 1
user1223
  • 197
  • Consider (1) Take $f=1_{[0,{\pi \over 2}]}$. (2) $f \in L^2[0,\pi]$. – copper.hat Mar 08 '18 at 21:27
  • @copper.hat Thanks! I know that when $f \in L^2[0,\pi]$ then the series is converge but how to explain that if we take function from $L_1$ then the series is diverges? – user1223 Mar 08 '18 at 21:30
  • If $f$ is also in $L^2$ then the coefficients are square summable. If the coefficients of $f$ are square summable, then $f$ is in $L^2$. So, if $f$ is in $L^1$ then the coefficients are square summable iff $f$ is also in $L^2$. – copper.hat Mar 08 '18 at 23:28

0 Answers0