Timeline for Interpret Fourier transform as limit of Fourier series
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 24, 2016 at 5:16 | comment | added | Nik Weaver | $\|P_n - I\| \to 0$ is norm convergence. $P_nf \to f$ for all $f$ is strong operator convergence. | |
| Feb 24, 2016 at 3:15 | comment | added | Lao-tzu | What do you mean by ''converge strongly''? Do you mean the operator norm ||P_n-Id|| go to 0? But this only shows P_n f go to f in L^2. | |
| Feb 21, 2016 at 2:53 | comment | added | Lao-tzu | I had make this into an MO question here: mathoverflow.net/questions/231716/… | |
| Feb 21, 2016 at 2:18 | comment | added | Lao-tzu | I think to prove that $P_{r_n} \chi_E \to \chi_E$ (a.e. pointwise or in $L^2$ sense), where $r_n\to\infty$ as $n\to \infty$, and $E$ is any Lebesgue measurable set with finite measure $|E|$, we need the following: $P_r \chi_E =\sum_{\alpha\in\lambda_r^*} 2\pi r |E\cap (\alpha+(-\frac{1}{4\pi r}, \frac{1}{4\pi r}))|\chi_{\alpha+(-\frac{1}{4\pi r}, \frac{1}{4\pi r})} \to \chi_E.$ I think it's easy when $E$ is an interval, but I can't prove it for general $E$. Could you please give some help? | |
| Jan 18, 2016 at 15:46 | comment | added | Nik Weaver | @Lao-tzu: multiplying by sinc takes the periodically extended $L^2(r\mathbb{R})$ into $L^2(\mathbb{R})$. Okay, $P_r$ if you like --- I meant the orthogonal projection onto the embedded $l^2(\frac{1}{2\pi r}\mathbb{Z})$ (but was thinking of taking a sequence of them). | |
| Jan 18, 2016 at 7:45 | comment | added | Lao-tzu | In the 2nd paragraph, do you mean that we extend a function $f \in L^2(r\mathbb{T})$ with its periodic extension to $\mathbb{R}$ and view the extended one as a tempered distribution on $\mathbb{R}$? But if so, the extended one doesn't lie in $L^2(\mathbb{R})$, so how to get an isometric embeddings of $L^2(r\mathbb{T})$ into $L^2(\mathbb{R})$ in line -5? Could you explain what is '$P_n$'? I think you may mean '$P_r$' (depends on the parameter $r$). | |
| Jan 16, 2016 at 3:10 | history | answered | Nik Weaver | CC BY-SA 3.0 |