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I am wondering if the following statements are correct:

(1) zero-order B-splines interpolation is equivalent to nearest-neighbor interpolation. $C^0$ continuity thus is not differentiable.

(2) first-order B-splines interpolation is equivalent to linear interpolation. $C^1$ continuity.

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In B-spline terminology, order means degree + 1. Therefore, a degree 3 B-spline has order 4. I am not sure what you meant by first-order B-spline exactly, but I am going to assume you are talking about degree 1 B-spline (as B-spline of order 1 means degree 0, which does not make too much sense). Degree 1 B-spline interpolation is indeed the same as linear interpolation. But it only has $C^0$ continuity.

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    $\begingroup$ A spline with order 1 (degree 0) is a piecewise constant function. So it's not continuous (i.e. not $C_0$). $\endgroup$
    – bubba
    Commented Apr 7, 2016 at 0:00
  • $\begingroup$ And it's not nearest-neighbour either (it only "looks behind" so to say). $\endgroup$
    – gerrit
    Commented Feb 3, 2017 at 12:21

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