Timeline for Seeking a class of functions for which sums approximate integrals well
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 21, 2014 at 22:12 | vote | accept | James Propp | ||
| Feb 21, 2014 at 21:21 | comment | added | Lucia | @James Propp: Also, if your function $f$ has reasonable decay, and you can carry out integration by parts twice, then you would get bounds on the Fourier transform that decay like $1/(1+|x|)^2$. This is of course analogous to your twice differentiable type condition; but as you can see above the Fourier condition is a bit more general ... . I imagine you were using Euler Maclaurin in your argument; the point of this answer is simply that you could also use Poisson sum. | |
| Feb 21, 2014 at 21:14 | comment | added | Lucia | Well three of your functions are very well known, and the fourth is easy enough to compute with. The Gaussian which is its own Fourier transform (up to scaling). The Fourier transform of $e^{-|t|}$ is $1/(1+x^2)$ (up to scaling again). The function $\max(1-|x|,0)$ is the Fejer kernel; Fourier transform $(\sin x/x)^2$ (up to scaling). The fourth function $\max (1- x^2, 0)$ needs a small computation using integration by parts (along the lines of calculating the Fejer kernel). I'm sure all would be found in a table; e.g. Gradshteyn & Ryzhik. | |
| Feb 21, 2014 at 20:24 | comment | added | James Propp | How can one see that the four listed functions fall into this category? Does one need to first derive explicit formulas for $\hat{f}$ in each case, or are more qualitative arguments available (possibly outsourcing all the hard work to results in the literature)? | |
| Feb 21, 2014 at 5:09 | history | answered | Lucia | CC BY-SA 3.0 |