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 Bspline has only finite support. How to see it using its Fourier series representation?
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 Bspline has only finite support. How to see it using its Fourier series representation? 


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 $c_j=({{e^{i\pi j/K}e^{i\pi j/K}}\over j/K})^p$. 


Since bsplines 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 bspline in terms of its Fourier transform. Centered scaled cardinal bsplines of order n have a closed form Fourier transform, ((sin omega)/omega)^{n1} up to scaling. 

