Let 0(x) denote the 5-spline function of Exercise 11.4.23. Assuming n > 4, let P" denote the vector space of periodic cubic splines based on the integer nodes xj = j for j = 0,... , n. (a) Prove that the B-splines

where m denotes the integer part of n/2. form a basis for Pn
(b) Graph the basis periodic 5-splines in the case n = 5.
(c) Let u(x) denote the periodic spline interpolant for the data values y0,... , yn-1. Explain how to write u(at) = a0β0 +--------+ αn - I βn-1 in terms of the 5 splines by solving a linear system for the coefficients ao..........a"_i.
(d) Write the periodic spline with y0 = y5 = 0, y1 - 2, y2 = 1, y3 = -1. y4 = -2, as a linear combination of the periodic basis 6- splines B0(x)..........54(jr). Plot the resulting periodic spline function

B, (x) ((x-j-m) mod n + m).