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Mar 27 at 6:53 history bounty ended Zhang Yuhan
Mar 20 at 8:01 comment added an_ordinary_mathematician @ZhangYuhan This case can be reduced to the previous one, see the edit
Mar 20 at 8:00 history edited an_ordinary_mathematician CC BY-SA 4.0
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Mar 20 at 6:53 history edited David Feldman CC BY-SA 4.0
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Mar 20 at 6:50 comment added Zhang Yuhan Thank you for the answser and the reference. You have proven that the positive dilations are linearly independent provided $f$ is non constant. However, if $f$ is an even function, then $f_{−1}-f_1=0$, and $(f_n)$ is linearly dependent.
Mar 20 at 6:46 comment added Zhang Yuhan I also found some posts regarding Fourier series of closed curves and added them to the question.
Mar 19 at 19:27 history undeleted an_ordinary_mathematician
Mar 19 at 19:27 history edited an_ordinary_mathematician CC BY-SA 4.0
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Mar 19 at 17:45 history deleted an_ordinary_mathematician via Vote
Mar 19 at 17:43 history answered an_ordinary_mathematician CC BY-SA 4.0