Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Does this concept in Fourier analysis which eliminates Gibbs' phenomenon and also introducing some new concepts, possibly have applications in any areas of physics, like QM or in any applied areas?

+ 1 like - 0 dislike
1234 views

This is more of seeking possible applications/contructs in physics for a new mathematical concept.

Main theme and motivation summary : 

----------


Motivation
----------
Consider a BV function $f$ with jumps, the Fourier partial integral function (analogous to Fourier partial sum in periodic case) as $\omega\to\infty$ does converge to $f$ pointwise except possibly at jumps. But the total variation of the partial integral function does not converge to the total variation of the function $f$, more over it goes to infinity. (attributed to Gibb's phenomenon). To overcome Gibb's phenomenon, alternate summation methods were suggested, like Cesaro summation but their convergence is very slow especially when $f$ has a jump.

In my approach, I propose a construction of a sequence of functions, just like partial sums, each one denoted as $P^f_{\omega}$ constructed using Fourier spectrum of $f$ only in the interval $(0,\omega)$. The function sequence is intended to converge to $f$ pointwise except at jump points, and not just that, but also overcome Gibb's phenomenon, there by variation of $P^f_{\omega}$ in any given open interval, converging to that of the function $f$ as $\omega\to\infty$.

More interesting part is that the function $P^f_{\omega}$ can possess jump discontinuties. (even when $f$ does not have jumps). I also predict that the convergence is much faster than Cesaro partial sums.

Another interesting part is that in Fourier analysis we try to approximate even functions that jump, with smooth functions. But here we use jumping functions to approximate jumping functions. (This problem could be formulated in Fourier series but I choose to do it for Fourier transform setup).

Please follow the MO link for the actual mathematics details : http://mathoverflow.net/q/208867/14414

asked Jun 14, 2015 in Mathematics by rajesh_d24 (5 points) [ revision history ]
edited Jun 15, 2015 by rajesh_d24

Hi Rajesh, welcome to PhysicsOverflow !

Maybe in addition to the MO link, you can give a short summary of the concepts you would like to see physics applications for, directly in this PO question too to help the reader?

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverfl$\varnothing$w
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...