B-splines 2.1 2003/04/23 2003/04/23 Alena Ixmocane Scott oetting@stat.rice.edu Alena Ixmocane Scott oetting@stat.rice.edu Sujesh Sreedharan sujesh@rice.edu Rebecca Willett willett@rice.edu Adan Galvan jago@rice.edu Charlet Reedstrom charlet@rice.edu B-splines basis splines This module explains B-splines, which form a basis for splines. B-splines are symmetrical, bell-shaped functions constructed from the ( n 1 ) fold convolution of a rectangular pulse β 0 x : β 0 x 1 x 0.5 0.5 x 0.5 0 β n x β + , 1 0 β + , 2 0 … β + , n 0 β + , n + 1 0 x Schoenberg Polynomial splines with equally spaced knots can be uniquely characterized in terms of a B-spline expansion. That is, for a spline s x of degree n, there exist unique coefficients c k such that s x k c k β n x k c β n x Understanding B-splines can help us better understand splines, since B-splines are the building blocks. Some properties of B-splines The derivative of a B-spline of order n with respect to x is equal to the sum of two B-splines of order n 1 evaluated at shifts of x x β n x β n - 1 x 0.5 β n - 1 x 0.5 The integral of a B-spline of order n is equal to the infinite sum of B-splines of order n 1 . t x β n t k 0 β n + 1 x 0.5 k Notice that this Fourier transform is related to the n 1 fold convolution construction of the B-splines. β ^ n ω ω 2 ω 2 n 1 B-splines are compactly supported. They are the shortest possible polynomial splines. β n x is a piecewise polynomial of degree n: β n x 1 n k 0 n 1 n 1 k 1 k x k n 1 2 n where x + n x n x 0 0 x 0