Central B-spline
Definition1
For $m\in \mathbb{N}$, the centered B-spline $B_{m}$ is defined as follows:
$$ B_{m}(x):= T_{-\frac{m}{2}}N_{m}(x)=N_{m}(x+{\textstyle \frac{1}{2}}) $$
Here, $T$ is the translation of the space $L^{2}$ space.
Description
It can also be defined as follows:
$$ B_{1}:= \chi_{[-1/2,1/2]},\quad B_{m+1}:=B_{m}*B_{1},\ m\in\mathbb{N} $$
Both definitions actually mean the same function. The key point here is that $B_{m}$ is defined to be an even function. As with the B-spline, it is easy to see that the following equation holds:
$$ B_{m+1}(x)=\int _{-\infty} ^{\infty} B_{m}(x-t)B_{1}(t)dt=\int_{-{\textstyle \frac{1}{2}}}^{{\textstyle \frac{1}{2}}}B_{m}(x-t)dt $$
The centered B-spline is merely a translation of the B-spline, and therefore has the same properties of the B-spline.
Properties
(a) $\mathrm{supp} B_{m}=[-\frac{m}{2},\frac{m}{2}]$
(b) $\displaystyle \int _{-\infty} ^{\infty} B_{m}(x)dx=1$
(c) For $m\ge 2$,
$$ \sum \limits_{k\in \mathbb{Z}}B_{m}(x-k)=1,\quad \forall x\in R $$
(c’) When $m=1$, the above equation holds for $x\in \mathbb{R}\setminus \left\{ \pm\frac{1}{2},\pm \frac{3}{2},\dots \right\}$.
(d) The Fourier transform of the centered B-spline is as follows:
$$ \widehat{B_{m}}(\gamma)=\left( \frac{e^{\pi i \gamma}-e^{-\pi i \gamma}}{2\pi i \gamma} \right)^{m}=\left( \frac{\sin (\pi\gamma)}{\pi \gamma} \right)^{m} $$
Here, the definition of the Fourier transform $\widehat{f}$ of $f$ is as follows:
$$ \widehat{f}(\gamma):=\int _{-\infty} ^{\infty} f(x)e^{-2\pi i x\gamma}dx $$
Proof
(a)
Since $\mathrm{supp}N_{m}=\left[ -\frac{m}{2}, \frac{m}{2} \right]$ and $B_{m}=T_{- \frac{m}{2}}N_{m}$,
$$ \mathrm{supp} B_{m} = \left[0-\frac{m}{2},m-\frac{m}{2} \right] = \left[ -\frac{m}{2},\frac{m}{2} \right] $$
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(b)
$$ \int _{-\infty} ^{\infty} N_{m}(x)dx=1 $$
and since $B_{m}=T_{- \frac{m}{2}}N_{m}$,
$$ \int _{-\infty} ^{\infty} B_{m}(x)dx=\int _{-\infty} ^{\infty} T_{-\frac{m}{2}}B_{m}(x)dx=1 $$
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(c)
$$ \sum \limits_{k \in \mathbb{Z}} N_{m}(x-k)=1,\quad \forall x\in \mathbb{R} $$
thus,
$$ \sum \limits_{k\in \mathbb{Z}}B_{m}(x-k)=\sum \limits_{k \in \mathbb{Z}} T_{-\frac{m}{2}}N_{m}(x-k)=1,\quad \forall x\in \mathbb{R} $$
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(c')
When $m=1$, if $N_{m}$ and $x\in \mathbb{R}\setminus \mathbb{Z}$ holds and $B_{m}=T_{- \frac{m}{2}}N_{m}$, then it holds for $x\in \mathbb{R}\setminus \left\{ \pm\frac{1}{2},\pm \frac{3}{2},\dots \right\}$.
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(d)
The Fourier transform of the B-spline is as follows:
$$ \widehat{N_{m}}(\gamma)=\left( \frac{1-e^{-2\pi i\gamma}}{2\pi i \gamma} \right)^{m} $$
Then, by the properties of the Fourier transform,
$$ \mathcal{F}\left[ N_{m}(x+\frac{m}{2}) \right] (\gamma)=e^{2\pi i \frac{ m}{2} \gamma}\widehat{N_{m}}(\gamma) $$
thus,
$$ \begin{align*} \widehat{B_{m}}(\gamma) =&\ \mathcal{F}\left[ B_{m}(x) \right] (\gamma) =\mathcal{F}\left[ N_{m}(x+\textstyle \frac{m}{2}) \right] (\gamma) \\ =&\ e^{2\pi i \frac{m}{2} \gamma}\widehat{N_{m}}(\gamma) \\ =&\ \left(e^{\pi i \gamma}\right)^{m}\left( \frac{1-e^{-2\pi i\gamma}}{2\pi i \gamma} \right)^{m} \\ =&\ \left( \frac{e^{\pi i \gamma}-e^{-\pi i \gamma}}{2\pi i \gamma} \right)^{m} \\ =&\ \left( \frac{\sin (\pi\gamma)}{\pi \gamma} \right)^{m} \end{align*} $$
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Ole Christensen, Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering (2010), p208-209 ↩︎