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I am currently trying to grasp how B-Spline functions work and seem to have hit an issue with its definition. One type of B-Spline basis function seems to be a cardinal B-Spline function, which I found in a textbook about this topic. There is an explicit definition given for a cardinal cubic B-Spline basis function with the following piecewise definition.

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Image source is the book https://link.springer.com/book/10.1007/978-3-658-29922-4 page 168, equation 9.1.1

Sadly the author does not explain how this definition came to be. So I am wondering how one could create such a function for different grades of polynomials.

For example how would the spline be defined for grade 4 or 5? I am looking for the pattern of the polynomials within its definition and maybe a short overview of the variants of B-Spline functions. The reason I am looking for such a function, is to get a B-Spline which can be differentiated more times than the cubic can be.

Remark: I did not study math, and already tried several online explainitions but could not make any sense of them or use them to create those polynomials.

Feirell
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  • You can get the definition from the wikipedia page https://en.wikipedia.org/wiki/B-spline#Cardinal_B-spline. Also I believe that you can define the uniform b-splines with convolution. If $u(x) = 1$ for $x\in (-1/2, 1/2)$ and $u(x) =0$ otherwise and $U_n= uu\ldotsu$ where the $u$ is repeated $n$ times, then $U_n$ is one of the b-splines of order $n$. means convolution. – irchans Jul 04 '23 at 13:46
  • @irchans I don't understand the definition the wikipedia article describes or exactly how to make use of your explaination. Could you please expand you commend to an answer, with explicit examples for the first four grades. This should probably make it much clearer! – Feirell Jul 04 '23 at 14:21

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