Can we Define a Basis on R^3 at Every Point Using Fractional Derivatives?

[Klaus_Hoffmann]

Let be y=f(x) a differentiable function, my question is if we can define a basis on R^3 at every point using $$\partial _{x}^{n}$$ n=0,1,2

For arbitrary 'n' even real numbers could be the same be defined using the fractional derivative to justify $$\partial _{x} ^{n} y(x)$$

So in every case the Wrosnkian is different from 0 except at several points, with this a possible purpose would be constructing a basis for a fractional-dimensional space to perform integration over R^{n} n being (positive) integer or real.

 

[matt grime]

A basis of what on R^3? (you wouldn't be the famous Klaus Hoffmann, musician and mellotron connosieur, would you?)

 

[tacman]

It is not possible to define a basis on R^3 at every point using fractional derivatives. A basis is a set of linearly independent vectors that span a vector space. In the case of R^3, we have a three-dimensional vector space, and a basis would consist of three linearly independent vectors. Fractional derivatives, on the other hand, are not defined for arbitrary 'n' even real numbers. They are only defined for positive integers or real numbers greater than or equal to 1. Therefore, it is not possible to use fractional derivatives to construct a basis in R^3 at every point.

Furthermore, the Wronskian, which is a determinant used to test for linear independence, would not be applicable in this context. The Wronskian is only defined for functions with integer order derivatives, not fractional derivatives. So even if we were able to define a basis using fractional derivatives, we would not be able to use the Wronskian to test for linear independence.

In addition, the purpose of constructing a basis in a fractional-dimensional space is unclear. Integration over R^n, where n is a positive integer or real number, is well-defined without the need for a fractional-dimensional basis. Therefore, there is no practical use for constructing a basis using fractional derivatives in this context.

In conclusion, it is not possible to define a basis on R^3 at every point using fractional derivatives, and there is no practical purpose for doing so. The concept of a basis is only applicable in the context of linear algebra, and fractional derivatives do not fit within this framework.