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

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SUMMARY

This discussion centers on the feasibility of defining a basis on R^3 at every point using fractional derivatives, specifically \partial _{x}^{n} for n=0,1,2. The conversation highlights that for arbitrary even real numbers, fractional derivatives can justify \partial _{x}^{n} y(x). It concludes that the Wronskian remains non-zero except at specific points, suggesting the potential for constructing a basis for fractional-dimensional spaces to facilitate integration over R^{n}, where n is a positive integer or real number.

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  • Understanding of fractional derivatives
  • Familiarity with the Wronskian determinant
  • Knowledge of basis concepts in linear algebra
  • Basic principles of integration in R^n
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This discussion is beneficial for mathematicians, physicists, and researchers interested in advanced calculus, fractional calculus, and the theoretical foundations of vector spaces.

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

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

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

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