Timeline for Is there a function that is not absolutely integrable in [−π,π] so that its Fourier Series Exists? [closed]
Current License: CC BY-SA 4.0
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| Jun 24, 2020 at 0:54 | history | closed |
Michael Renardy user44191 R.P. coudy Yemon Choi |
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| Jun 24, 2020 at 0:54 | comment | added | Yemon Choi | The question has also been posted on MathStackExchange math.stackexchange.com/questions/3730663/… which I think is a more appropriate place for it | |
| Jun 24, 2020 at 0:51 | comment | added | Yemon Choi | It is difficult to answer this question unless you supply a precise definition of what you mean by saying "the Fourier series exists". Given a function on $[-\pi,\pi]$, how do you propose to define its Fourier coefficients? And assuming you can define each individual Fourier coefficient, what do you mean by a series "existing"? Do you require the series to converge in some way? If not, then aren't you just asking if each Fourier coefficient exists? | |
| Jun 23, 2020 at 7:55 | comment | added | SAS | @PieroD'Ancona yes, I know. In this theory all is very nice because $\mathcal{L}_2$ is a Hilbert space. However, Serie Fourier also can be defined for another type of function (Function $f$ such that $f \notin \mathcal{L_2}$), and in these cases is this problem. | |
| Jun 23, 2020 at 2:20 | review | Close votes | |||
| Jun 24, 2020 at 0:54 | |||||
| Jun 22, 2020 at 22:46 | comment | added | SAS | I consider that this is false, because $\int_{a}^{\infty} \dfrac{sin(x)}{x}dx$ converges $\forall a>0$ but $\int_{a}^{\infty} \dfrac{|sin(x)|}{x} dx$ diverges | |
| Jun 22, 2020 at 22:43 | answer | added | Bazin | timeline score: 3 | |
| Jun 22, 2020 at 22:26 | comment | added | Dieter Kadelka | If $\int |f|d\mu = \infty$, then always $\int fd\mu = \pm \infty$ (if either $\int f^+d\mu < \infty$ or $\int f^-d\mu < \infty$) or this integral does not exist. | |
| Jun 22, 2020 at 22:18 | answer | added | Tanya Vladi | timeline score: 0 | |
| Jun 22, 2020 at 22:17 | comment | added | SAS | This is the problem, because if $\int_{-\pi}^{\pi} |f(x)|dx$ converges then Fourier Series of $f$ exists always , but is possible that is there a functión that $\int_{-\pi}^{\pi} |f(x)|dx$ diverges (no necessarily $\int_{-\pi}^{\pi} f(x)dx$ diverges) and its Fourier series exists? if it is not possible why? | |
| Jun 22, 2020 at 22:08 | comment | added | Dieter Kadelka | How do you define $\int_{-\pi}^\pi f(x)dx$, if $\int |f|dx = \infty$? More concrete: What is then its Fourier Series? | |
| Jun 22, 2020 at 21:49 | review | First posts | |||
| Jun 22, 2020 at 22:25 | |||||
| Jun 22, 2020 at 21:48 | history | asked | SAS | CC BY-SA 4.0 |