Starting with the equations of our two spline segments and their derivitives we can construct the two desired spline segments.

The resulting family of reconstruction filters is:

These bicubic reconstruction splines are commonly called

Mitchell's 2-parameter spline family

Notes:

k(0)=(3 - B)/3,   k(1)=B/6
thus, if B = 0 then k(0) = 1, k(1) = 0 like sinc(x)