Timeline for Bound of Coefficients of Fourier Series of Composition
Current License: CC BY-SA 4.0
20 events
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| Aug 10, 2019 at 4:10 | history | edited | Halbort | CC BY-SA 4.0 |
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| Aug 10, 2019 at 4:09 | comment | added | Halbort | Yes but this does not improve the bound for repeatedly composed functions. | |
| Aug 10, 2019 at 4:07 | comment | added | reuns | The trick for the Hölder continuity is $F_n = \int_0^{2\pi} f(x) e^{-inx}dx =\int_0^{2\pi} f(x+\pi / n) e^{-in(x+\pi /n)}dx=- \int_0^{2\pi} f(x+\pi / n) e^{-inx}dx$, $ 2F_n = \int_0^{2\pi} (f(x)-f(x+\pi /n)) e^{-inx}dx$. If $|f(x)-f(x+y)| \le C y^a$ then $|F_n| \le 2 \pi C n^{1-a}$. I have explained why for the decay your function is just any $C^k$ function | |
| Aug 10, 2019 at 4:05 | comment | added | Halbort | Yes I know that. Is there anyhting better you can say about the Fourier coefficients of $f(f(x))$ or $f^k(x)$ in general. | |
| Aug 10, 2019 at 3:48 | history | edited | Halbort | CC BY-SA 4.0 |
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| Aug 10, 2019 at 3:44 | comment | added | Halbort | Can you give a bound if they are 1-Holder continuous? What if they are $\alpha$-Holder Continuous for $\alpha > 0$ | |
| Aug 10, 2019 at 3:38 | comment | added | reuns | A continuous periodic function always is uniformly continuous. | |
| Aug 10, 2019 at 3:35 | comment | added | Halbort | What if both functions are uniformly continuous. | |
| Aug 10, 2019 at 3:35 | history | edited | Halbort | CC BY-SA 4.0 |
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| Aug 10, 2019 at 3:32 | history | edited | user64494 | CC BY-SA 4.0 |
The title is improved.
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| Aug 10, 2019 at 3:24 | comment | added | reuns | The coefficients of $h \in C^k$ are $o(n^{-k})$ and $\sum_n |H_n|^2 n^{2k} < \infty$. To get stronger bounds you need things like $a$-Hölder continuity of $h^{(k)}$ | |
| Aug 10, 2019 at 3:19 | comment | added | Halbort | I have added the negative terms. I had forgotten to. Is there some way to give a better bound like $O(n^{-k + \frac{1}{\log(n)}})$ I want to show that the decay of $f^p(x)$ is slower. | |
| Aug 10, 2019 at 3:17 | history | edited | Halbort | CC BY-SA 4.0 |
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| Aug 10, 2019 at 3:05 | comment | added | reuns | $f$ is real valued iff its Fourier coefficients satisfy $F_{-n} = \overline{F_n}$. Also the decay of the Fourier coefficients depends on the local smoothnesses, and locally the composition of $C^k$ functions surjects on the $C^k$ functions. | |
| Aug 10, 2019 at 2:56 | comment | added | Halbort | I edited the question. | |
| Aug 10, 2019 at 2:55 | history | edited | Halbort | CC BY-SA 4.0 |
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| Aug 10, 2019 at 2:40 | comment | added | reuns | What do you mean with $f \circ g$ for $f$ defined on $\Bbb{R}$ or $\Im(x)\ge 0$ and $g$ complex valued | |
| Aug 10, 2019 at 1:35 | history | edited | Halbort | CC BY-SA 4.0 |
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| Aug 9, 2019 at 21:55 | history | edited | Halbort | CC BY-SA 4.0 |
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| Aug 9, 2019 at 21:46 | history | asked | Halbort | CC BY-SA 4.0 |