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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?

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  • $\begingroup$ Usually the basis elements of some vector space are also elements of that vector space. $\endgroup$ Aug 7, 2023 at 9:34

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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 :)

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  • $\begingroup$ 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? $\endgroup$
    – Ronald
    Aug 6, 2023 at 12:34
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    $\begingroup$ 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. $\endgroup$
    – rafexiap
    Aug 6, 2023 at 12:35
  • $\begingroup$ 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? $\endgroup$
    – Ronald
    Aug 6, 2023 at 12:42
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    $\begingroup$ 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. $\endgroup$
    – rafexiap
    Aug 6, 2023 at 12:54
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    $\begingroup$ This just proves my point: people use incorrect terminology all the time. $\endgroup$
    – rafexiap
    Aug 6, 2023 at 13:18

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