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How to approximate a function using B-spline interpolation?

B splines are already built into MMA. To expand Sin[x] over the region 0 o 2 Pi into B splines of degree d=3 we choose a basis of b splines with uniform knots at a distance of 1/(d+1): ...
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Understanding Interpolation with Cubic Splines

We can confirm that the Interpolation produces a "not-a-knot" spline when using the "Spline" method by comparing the results of different spline ...
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