@MfA,
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and also continuity of first and second derivatives, where the second derivatives of the two endpoints are zero.
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Derivative of the interpolation kernel of spline 16 goes from -4/5 to -7/15 at the crossover ... not very continuous.
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No, you are referring to the blending polynomials here, their derivatives are indeed not continuous.
The derivatives of the interpolating functions (S(x) and the other ones) are continuous at the crossovers. Well, in case you have a multiple of them (for k>2). Note that only one of them is used (namely the one for [0,1]) for interpolating pixels. The source pixels are set to (y0, ..., y(k-1)).