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.
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.