4.5.41 Problem

Suppose that fk4(a,b) and SΔ(x) are spline interpolating functions that interpolate node points Δ={a=x0<x1<<xn=b}. Show that:

fSΔ2=ab(f(x)SΔ(x))f(4)(x)dx

if only one of the following conditions is satisfied:

  1. f(x)=SΔ(x)x=a,b

  2. f(x)=SΔ(x)x=a,b

  3. fkp4(a,b),periodicSΔ

Solution:

fSΔ2=ab|f(x)SΔ(x)|2dx=ab(f(x)SΔ(x))(f(x)SΔ(x))dx=ab(f(x)SΔ(x))f(x)dxab(f(x)SΔ(x))SΔ(x)dx:=I

If we apply integration by parts twice on each of above integrals, we will have:

I=[(f(x)SΔ(x))f(x)]ab[(f(x)SΔ(x))f(x)]ab+ab(f(x)SΔ(x))f(4)(x)dx[(f(x)SΔ(x))SΔ(x)]ab[(f(x)SΔ(x))SΔ(x)]xi1+xii=1nxi1xi(f(x)SΔ(x))Sδ(4)(x)dx=[(f(x)SΔ(x))(f(x)SΔ(x))]ab[(f(x)SΔ(x))f(x)]ab+ab(f(x)SΔ(x))f(

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