Let
a,bC_(Delta )^(1)[a,b]:={finC[a,b]:f|_([x_(i),x_(i+1)])inC^(1)[x_(i),x_(i+1)] for iin{0,1,dots,n-1}}f|_([x_(i),x_(i+1)])x_(i),x_(i+1)sf_(i)inRx_(i),iin{0,dots,n}||f^(')-s^(')||_(2)^(2)=||f^(')||_(2)^(2)-||s^(')||_(2)^(2)finC_(Delta )^(1)[a,b]f(x_(i))=f_(i)||f^(')-s^(')||_(2)^(2)=min_(psi inS_(Delta ,1))||f^(')-psi ^(')||_(2)^(2)V:={finC_(Delta )^(1)[a,b]:f(x_(i))=f_(i)AAi}sV->R,f|->||f^(')||_(2)||f||_(2)=(int_a^b f(x)^(2)dx)^((1)/(2))int_a^b [f^(')(x)-s^(')(x)]s^(')(x)dx
Let
={a=xo<x1<...<xn=b}
be a decomposition of the interval [a, b] and
C}[a,b]:={f eC[a,b]: f|[x,x+] EC1[xi,xi+1]for ie{0,1,...,n-1}}
be the space of continuous, piecewise differentiable functions. Here fl[x;,++] denotes the restriction to [xi, Xi+1]. Let s be the linear spline that interpolates given data fi E R at xi, i {0,...,n}.
(a) Show that |I f' - s'lI3 = Il f'lI3 -- I|s' ll3 for all f E C}[a, b] with f(xi) = fi.
(b) Demonstrate that
IIf' -s'lI3= min IIf'-y'II3 WES.1
(c) Let V := {f E C}[a, b] : f(xi) = fi V i} be the space of continuous, piecewise differentiable functions interpolating the data. Show that the linear interpolating spline s is the minimiser of
V- R,f H> IIf'll2
[a[f'(x)-s'(x)]s'(x)dx.
