fourier series of b-spline - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:26:54Z http://mathoverflow.net/feeds/question/27941 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27941/fourier-series-of-b-spline fourier series of b-spline vilvarin 2010-06-12T16:32:16Z 2010-06-12T22:10:31Z <p>Fourier series of a function (B spline) are given by: $$s(x)=\sum_{j=-\infty}^{\infty}\operatorname{sinc}\Bigl[\pi\frac{j}{K}\Bigr]^{p}\exp[2\pi ijx]$$</p> <p>But B-spline has only finite support. How to see it using its Fourier series representation?</p> http://mathoverflow.net/questions/27941/fourier-series-of-b-spline/27969#27969 Answer by coudy for fourier series of b-spline coudy 2010-06-12T20:09:23Z 2010-06-12T20:09:23Z <p>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. </p> <p>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 $1\over j$.</p> <p>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 $e^{i\pi ja}\over j^k$, $k\in \mathbb{N},a\in \mathbb{R}$. </p> <p>You can see that this is the case in your formula by expanding the coefficient $c_j=({{e^{i\pi j/K}-e^{i\pi j/K}}\over j/K})^p$.</p>