Anupam Saxena Associate Professor Indian Institute of Technology KANPUR

(tj  t)m1+(tj  t)m1+ tjtj t f(t j ; t) = (t j  t) m  1 + is continuous at t = t j and so are its derivatives up to m  2 (t j  t) m  1 + is a spline of order m An nth order (n  1 degree) spline is C n-2 continuous

Since each truncated power function is a spline, their weighted linear combination will be a spline of order m treat t as constant and compute the m th divided difference using s = t i  m, t i  m+1, …, t i, mth divided difference of the truncated function f(s; t) = (s  t) m  1 + B  spline basis function w(t) = (t  t i  m )(t  t i  m+1 )… (t  t i ) and w ’ (t) = dw/dt

t > tit > ti  (t) = 0 (t i+r  m  t) m  1 +, r = 0, …, m are all zero t  t i-m  (t) = 0  (t) is the m th divided difference of a pure (m  1) degree polynomial  (t) is standardized Peano’s theorem g(t) = t m, g m (t) = (m)! mth divided difference of t m is 1 An nth order (n  1 degree) spline is C n-2 continuous

divided differences of the product of two functions, h(t) = f(t)g(t) h[t 0, t 1,…,t k ] = f[t 0, t 1, …,t r ] g[t r, t r+1, …,t k ] = f[t 0 ]g[t 0, t 1, …,t k ] + f[t 0, t 1 ]g[t 1, …,t k ] + … + f[t 0, t 1, …,t k  1 ]g[t k  1,t k ] + f[t 0, t 1, …,t k ] g[t k ] h k (t j ; t) = (t j  t) k  1 + = (t j  t) k  2 + (t j  t) + = h k  1 (t j ; t) (t j  t) +

h k [t i  k,..., t i ;t ] = h k  1 [t i  k,…, t i  1 ;t] + h k  1 [t i  k,…, t i ;t](t i  t) k th divided difference of (t j  t) k  1 + : B-spline M k, i (t) M k-1,i-1 (t)

h k [t i  k,..., t i ;t ] = h k  1 [t i  k,…, t i  1 ;t] + h k  1 [t i  k,…, t i ;t](t i  t) M k, i (t)M k  1,i  1 (t) + {h k  1 [t i  k+1,…, t i ;t ]  h k  1 [t i  k,…, t i  1 ;t]} = {M k  1, i (t)  M k  1, i  1 (t)} M k, i (t) + = M k  1,i  1 (t) M k, i (t) = M k  1,i  1 (t) + M k  1, i (t) similar to the de Casteljau’s algorithm repeated linear interpolation is performed between two consecutive splines a table to construct splines may also be generated