Subtlety about the definition of B-splines

Question

I came across the following definition for the zero'th order B-spline

$$b_0(x) = \left\{ \begin{array}{lr} 0 & |x|>1/2\\ 1 & |x|<1/2\\ 1/2& |x|=1/2. \end{array} \right. $$

Why does $b_0$ have to assume the value 1/2 in $x=1/2$?

Answer 1

One might combine several of these box functions as in $b_0(x)+b_0(x-1)$ and expect a constant function on $(-\frac12,\frac32)$ as a result. For that you need $b_0(-\tfrac12)=1-b_0(\tfrac12)$. If symmetry is sought, one has to take equal values, which only leaves $\tfrac12$ as the value at the jumps.

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One other preference for this value I can see comes from compatibility with the Fourier transform/series, where function values at jumps are the midpoints of the one-sided limits.

But even that is a stretch since the inverse transform of the $\operatorname{sinc}$ function does not exist as pointwise defined integral.

However, it does work with periodic rectangular pulses, where the Fourier series converges pointwise if this midpoint value is used.

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A periodic function $f$ with left and right limits everywhere for function and first derivative values has a Fourier series that converges pointwise to $$s(x)=\tfrac12(f(x+0)+f(x-0)),$$ where $f(x\pm 0)=\lim_{h>0,\;h\to 0}f(x\pm h)$. Thus if the original function already satisfies $$f(x)=\tfrac12(f(x+0)+f(x-0)),$$ which is non-trivial only at jumps, then it is exactly reproduced by the Fourier series.

Answer 2

You don't have to define $b_0(1/2) = 1/2$, and in fact the usual definition does not assume this. See this page, for example.

But, I don't think it's worth worrying about the precise definition of the zero-th order basis functions, anyway. You don't get anything very interesting until you apply the first recursive step, and generate the linear (degree 1) basis functions. In fact, it might even be reasonable to start the recursive definition from degree = 1. The degree 1 basis functions are "hat" functions, and at least they are continuous, assuming distinct knots. Discontinuous b-splines are not very useful.