# 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?

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

Main theme and motivation summary :

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Motivation
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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).

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