Reproducing Polynomials with B-Splines

A B-Spline of order $N$ is known to be able to reproduce any polynomial up to order $N$:

$\sum _{n\in \mathbb {Z} }c_{m,n}\beta _{N}(t-n)=t^{m}$

In words, a proper linear combination of shifted versions of a B-Spline can reproduce any polynomial up to order $N$. This is needed for different applications, for example, for the Sampling at Finite Rate of Innovation (FRI) framework. In this case any kernel $\varphi$ reproducing polynomials (that is, satisfying the Strang-Fix conditions) can be used. However, among all possible kernels, the B-Splines have the smallest possible support.

An important question is how to obtain the coefficients $c_{m,n}$ for the reproduction-formula. In this small article, I describe one way.

Starting from

$\sum _{n\in \mathbb {Z} }c_{m,n}\varphi (t-n)=t^{m}$

the coefficients can be obtained using the dual of $\varphi$, ${\tilde {\varphi }}$ (I set $\beta _{N}=\varphi$ for consistency with my notes):

$c_{m,n}=\int _{-\infty }^{\infty }t^{m}{\tilde {\varphi }}(t-n)\,dt$

However, even if the dual would be known, solving the infinite integral is only feasible when the dual has finite support. This is the case with the B-Spline itself but not with its dual!

A closer look at the formula tells that this is nothing more than a convolution (under the assumption that ${\tilde {\varphi }}$ is symmetric which is the case):

$c_{m,n}=\int t^{m}{\tilde {\varphi }}(-(n-t))\,dt=\int t^{m}{\tilde {\varphi }}(n-t)\,dt=(t^{m}*{\tilde {\varphi }})(n)$

Now, this can be transformed to fourier domain:

$(t^{m}*{\tilde {\varphi }})(n)={\mathcal {F}}^{-1}\left\{{\mathcal {F}}\left\{t^{m}\right\}{\tilde {\Phi }}(\omega )\right\}={\mathcal {F}}^{-1}\left\{j^{m}{\sqrt {2\pi }}\delta ^{(n)}(\omega ){\tilde {\Phi }}(\omega )\right\}=j^{m}{\sqrt {2\pi }}{\mathcal {F}}^{-1}\left\{\delta ^{(n)}(\omega ){\tilde {\Phi }}(\omega )\right\}$

Writing the inverse of this expression yields:

$j^{m}{\sqrt {2\pi }}{\frac {1}{\sqrt {2\pi }}}\int _{-\pi }^{\pi }\delta ^{(n)}(\omega ){\tilde {\Phi }}(\omega )e^{j\omega n}\,d\omega =j^{m}\int _{-\infty }^{\infty }\delta ^{(n)}(\omega )\underbrace {{\tilde {\Phi }}(\omega )e^{j\omega n}} _{f(\omega )}\,d\omega$

It is known that:

$\int \delta ^{(n)}(x)f(x)\,dx=(-1)^{n}f^{(n)}(0)$

so that

$j^{m}\int _{-\infty }^{\infty }\delta ^{(n)}(\omega )f(\omega )\,d\omega =j^{m}(-1)^{m}\left.{\frac {\partial ^{m}}{\partial \omega ^{m}}}f(\omega )\right|_{\omega =0}$

Now the whole procedure has been reduced to calculate the derivative of $f(\omega )$ and set the result to zero.

An open question is how to obtain the dual of $\varphi$. As the reproduction formula spans a vector space, the $\varphi$ must be at least bi-orthogonal to ${\tilde {\varphi }}$. This translates in fourier domain to:

${\tilde {\Phi }}(\omega )={\frac {\Phi (\omega )}{\sum _{k\in \mathbb {Z} }|\Phi (\omega +2\pi k)|^{2}}}$

The fourier transform of a B-Spline of order $N$ is (e.g. ):

$\mathrm {B} _{N}(\omega )=\Phi (\omega )=\left({\frac {\sin(\omega /2)}{\omega /2}}\right)^{N+1}=\mathrm {sinc} ^{N+1}(\omega /2)$

The following derivation of the sum is borrowed from . For this derivation to work, I set $L=N+1$ temprarily:

$\sum _{k\in \mathbb {Z} }|\Phi (\omega +2\pi k)|^{2}=\sum _{k\in \mathbb {Z} }\left|\mathrm {sinc} \left({\frac {1}{2}}(\omega +2\pi k)\right)^{L}\right|^{2}=\sum _{k\in \mathbb {Z} }\left|\mathrm {sinc} \left({\frac {1}{2}}(\omega +2\pi k)\right)\right|^{2L}$

and because $2L$ is always even:

$=\sum _{k\in \mathbb {Z} }{\frac {\sin ^{2L}({\frac {1}{2}}(\omega +2\pi k))}{\left({\frac {1}{2}}(\omega +2\pi k)\right)^{2L}}}=\sum _{k\in \mathbb {Z} }{\frac {\sin ^{2L}({\frac {\omega }{2}}+\pi k))}{({\frac {\omega }{2}}+\pi k)^{2L}}}$

Because of the periodicity it is known that

$\sin ^{2L}(x+\pi k)=\sin ^{2L}(x)$

such that

$=\sin ^{2L}({\frac {\omega }{2}})\sum _{k\in \mathbb {Z} }{\frac {1}{({\frac {\omega }{2}}+\pi k)^{2L}}}$

And finally the following relation is used:

$\sum _{k}{\frac {1}{(x+\pi k)^{2L}}}=-{\frac {1}{(2L-1)!}}{\frac {d^{2L-1}}{dx^{2L-1}}}\cot {x}$

in order to finally obtain:

$\sum _{k\in \mathbb {Z} }\left|\mathrm {sinc} \left({\frac {1}{2}}(\omega +2\pi k)\right)^{L}\right|^{2}=-\sin ^{2L}\left({\frac {\omega }{2}}\right){\frac {1}{(2L-1)!}}{\frac {d^{2L-1}}{d\left({\frac {\omega }{2}}\right)^{2L-1}}}\cot {\left({\frac {\omega }{2}}\right)}$

and with $L=N+1$:

$\sum _{k\in \mathbb {Z} }|\Phi (\omega +2\pi k)|^{2}=-\sin ^{2(N+1)}\left({\frac {\omega }{2}}\right){\frac {1}{(2N+1)!}}{\frac {d^{2N+1}}{d\left({\frac {\omega }{2}}\right)^{2N+1}}}\cot {\left({\frac {\omega }{2}}\right)}$

Therefore, together with $\Phi (\omega )$ this yields:

${\tilde {\Phi }}(\omega )={\frac {(2N+1)!}{\left({\frac {\omega }{2}}\right)\sin \left({\frac {\omega }{2}}\right)^{N+1}{\frac {d^{2N+1}}{d\left({\frac {\omega }{2}}\right)^{2N+1}}}\cot {\left({\frac {\omega }{2}}\right)}}}$

and finally substituting for $t(\omega )$:

$f(\omega )={\frac {(2N+1)!}{\left({\frac {\omega }{2}}\right)\sin \left({\frac {\omega }{2}}\right)^{N+1}{\frac {d^{2N+1}}{d\left({\frac {\omega }{2}}\right)^{2N+1}}}\cot {\left({\frac {\omega }{2}}\right)}}}e^{j\omega n}$

As this function is not well defined it is better to use the limit:

$c_{m,n}=j^{m}\lim _{\omega \rightarrow 0}f(\omega )$

Examples for a cubic spline

For a cubic spline (N=3) the coefficients are:

\begin{array}{lcr} c_{0,n} & = & 1 \\ c_{1,n} & = & n \\ c_{2,n} & = & \frac{1}{3}\left( -1 + 3n^2 \right) \\ c_{3,n} & = & -n + n^3 \end{array}