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Suppose I have two arrays of real numbers $x[i]$, $y[i]$ which can be considered as some smooth 2D curve points.

How can I find array of its curvature approximate values in each node $k[i]$?

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There have been papers written on this topic. Here is one, which can lead you to others:

Anoshkina, Elena V., Alexander G. Belyaev, and Hans-Peter Seidel. "Asymtotic Analysis of Three-Point Approximations of Vertex Normals and Curvatures." VMV. 2002.


         
          Figure from Belyaev, "A note on invariant three-point curvature approximations (...)." (1999).


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