     suivant: Interpolation with spline functions monter: Natural splines: spline précédent: Definition   Table des matières   Index

### Theorem

The set of spline functions of degree l on is a -vectorial subspace of dimension n + l.

Proof
On [a, x1], s is a polynomial A of degree less or equal to l, hence on [a, x1], s = A(x) = a0 + a1x + ...alxl and A is a linear combinaison of 1, x,...xl.
On [x1, x2], s is a polynomial B of degree less or equal to l, hence on [x1, x2], s = B(x) = b0 + b1x + ...blxl.
s has continuous derivatives up to order l - 1, hence : 0 j l - 1,    B(j)(x1) - A(j)(x1) = 0

therefore B(x) - A(x) = (x - x1)l or B(x) = A(x) + (x - x1)l.
Define the function :

q1(x) =  Hence :

s|[a, x2] = a0 + a1x + ...alxl + q1(x)

On [x2, x3], s is a polynomial C of degree less or equal than l, hence on [x2, x3], s = C(x) = c0 + c1x + ...clxl.
s has continuous derivatives until l - 1, hence : 0 j l - 1,    C(j)(x2) - B(j)(x2) = 0

therefore C(x) - B(x) = (x - x2)l or C(x) = B(x) + (x - x2)l.
Define the function :

q2(x) =  Hence : s|[a, x3] = a0 + a1x + ...alxl + q1(x) + q2(x)
And so on, the functions are defined by : 1 j n - 1,qj(x) =  hence,

s|[a, b] = a0 + a1x + ...alxl + q1(x) + .... + qn-1(x)

and s is a linear combination of n + l independant functions 1, x,..xl, q1,..qn-1.     suivant: Interpolation with spline functions monter: Natural splines: spline précédent: Definition   Table des matières   Index
giac documentation written by Renée De Graeve and Bernard Parisse