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12. Local Operators
 
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Despite of some interesting results, we have left unanswered important questions about the functions of the derivative. For instance, it would be very interesting to know if integer order derivatives are or not the only kind of functions of the derivative that are local, and if are or not other kind of local operators that are not functions of the derivative.

Looking at (5.2) it seems that in fact the only local functions of the derivative are integer order derivatives and their finite sums, being non-locality the result of adding infinite terms with displacement functions steadily increasing the distance to the point in which the function is evaluated. However, this impediment can be overcome by defining the derivative in the following way

(12.1)

so that the limit assures locality. In order to avoid another feature of fractional derivatives that sometimes is not wanted -the fact that the derivative of a constant is not zero- it is usual to define the local fractional derivative as

(12.2)

or similar forms. So there can be operators being local and taking an infinite number of evaluations of the function, provided that the points in which these evaluations are carried out -or the value of their displacement function- remains being infinitesimal. The question now is if there are other local operators besides the local fractional derivative and functions of it.

It could be argued that if a function admits its expansion in Taylor series, all the information about the function is indeed in its derivatives, so that any operator giving information about the function will be unnecessary and reducible to a sum of integer order derivatives. However, on the one hand there are functions that are not differentiable but are local fractional differentiable, and on the other hand, even if local operators could be reduced to ordinary derivatives for differentiable functions, that does not assure that ordinary derivatives are the most appropriate local operators for all the tasks.

Characterizing all local operators is important, for locality makes of them tools that could be useful in understanding Nature.