Fourier analysis nonlocality

Now, thanks to the development of the local wavelet analysis [3], it is possible to represent mathematically, according to the observations, finite waves with a well-defined energy. Therefore it is possible to represent the 0 wave, the extended yet localized part of the particle, with a defined energy by a wavelet. The wavelets, or finite waves, were developed in geophysics by Morlet in the early 1980s, to avoid some shortcomings of the nonlocal Fourier analysis. 

The usual uncertainty relations are a direct mathematical consequence of the nonlocal Fourier analysis therefore, because of this fact, they have necessarily nonlocal physical nature. In this picture, in order to have a particle with a well-defined velocity, it is necessary that the particle somehow occupy equally all space and time, meaning that the particle is potentially everywhere without beginning nor end. If, on the contrary, the particle is perfectly localized, all infinite harmonic plane waves interfere in such way that the interference is constructive in only one single region that is mathematically represented by a Dirac delta function. This implies that it is necessary to use all waves with velocities varying from minus infinity to plus infinity. Therefore it follows that a well-localized particle has all possible velocities. 

If, instead of the nonlocal Fourier analysis, one uses the local wavelet analysis to represent a quantum particle, the uncertainty relationships may change in form. On the other hand, this process has the advantage of containing the usual uncertainty relations when the size of the basic gaussian wavelet increases indefinitely. 

It is easily seen from (90) that when the size of the basic wavelet Axo is large enough, the new relation turns itself into the old, usual Heisenberg relations, which is a very satisfactory result. This situation corresponds to the limiting case when the wavelet analysis transforms into the nonlocal Fourier analysis. 

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