The piecewise cubic spline should satisfy the following:

1. The function must be symmetric:
   
   k(x) = k(-x)
   
   We can reformulate our general cubic such that it is symmetric as follows: This also allows us to consider only 2 segments instead of 4.
   
2. C₀ and C₁ continuity between spline segments
   
   k₁(1) = k₂(1),   k₂(2) = 0
   k'₁(0) = 0,   k'₁(1) = k'₂(1),   k'₂(2) = 0
3. In order to assure that a uniform array of samples is reconstructed as a uniform continuous function it should satisfy the following convex relationship: