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When you write a spline curve as a linear combination of b-spline basis functions, it's called a "b-spline". The basis functions are generated recursively by the deBoor-Cox algorithm, starting with degree zero and working upwards to higher degrees. These b-spline basis functions are thus written in terms of (i.e., as a linear combination of) other b-spline basis functions, so they are also b-splines?

Ronald
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"B" in b-splines stands for basis. Unfortunately many people incorrectly refer to other splines (=piecewise-polynomials) as b-splines, and to b-splines as "basis b-splines" or "b-spline basis functions". Do not become one of those people :)

rafexiap
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  • Now I'm even more confused. We try to achieve a spline that is a linear combination of b-spline basis functions, right? That spline is called a b-spline. But are the b-spline basis functions also a b-spline? – Ronald Aug 06 '23 at 12:34
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    No. A non-trivial linear combination of B (basis) splines is [correctly] called a spline. The only b-splines are basis splines. Unfortunately people use incorrect terminology all the time. – rafexiap Aug 06 '23 at 12:35
  • But you can have splines that are not formed based on a linear combination of B (basis) splines, right? Isn't it then wrong to say that they are also "splines", based on your definition? – Ronald Aug 06 '23 at 12:42
  • Could you give me an example of a spline, that cannot be written as a combination of b-splines? – rafexiap Aug 06 '23 at 12:44
  • What I meant is that they can be written, but they are not. They are written in another way. – Ronald Aug 06 '23 at 12:48
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    The definition of a spline is that it is a piecewise polynomial. Any spline can be written as a combination of B-splines. I am not sure how this can be made more clear. – rafexiap Aug 06 '23 at 12:54
  • So you are saying that this answer from @bubba is wrong? https://math.stackexchange.com/a/700183/744941 "When you write a spline curve as a linear combination of b-spline basis functions in this way, it's called a "b-spline" – Ronald Aug 06 '23 at 13:12
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    This just proves my point: people use incorrect terminology all the time. – rafexiap Aug 06 '23 at 13:18