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Fourier series of a function (B spline) are given by: $$s(x)=\sum_{j=-\infty}^{\infty}\operatorname{sinc}\Bigl[\pi\frac{j}{K}\Bigr]^{p}\exp[2\pi ijx]$$

But B-spline has only finite support. How to see it using its Fourier series representation?

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up vote 4 down vote accepted

A function is a piecewise polynomial if and only f it is a linear combination of functions, each of them having some derivative equal to a finite sum of Dirac measures.

The jth Fourier coefficient of the Dirac at a is $e^{2\pi ija}$, and integrating amounts to multiplying the coefficient by some power of $1\over j$.

As a result, a function is a spline with finite support if and only if its Fourier coefficients $c_j$ can be written as a finite linear combination of terms of the form $e^{i\pi ja}\over j^k$, $k\in \mathbb{N},a\in \mathbb{R}$.

You can see that this is the case in your formula by expanding the coefficient $c_j=({{e^{i\pi j/K}-e^{i\pi j/K}}\over j/K})^p$.

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Since b-splines have finite support, you can extend them to be periodic. This makes it possible to apply the Poisson summation formula to describe the Fourier series expansion of a b-spline in terms of its Fourier transform. Centered scaled cardinal b-splines of order n have a closed form Fourier transform, ((sin omega)/omega)^{n-1} up to scaling.

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