Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 5

Suppose first that $f-u^{\ast }$ is orthogonal to ${\mathcal{U}}_n$. Every $u\in {\mathcal{U}}_n$ can be written $u=u^{\ast }+v$ with $v\in {\mathcal{U}}_n$. The Pythagorean theorem in an inner-product space givesThus $u^{\ast }$ is a best approximation.

Conversely, if $u^{\ast }$ minimizes the distance, then for every $v\in {\mathcal{U}}_n$ the quadratichas its minimum at $t=0$. Differentiating there gives $q^{\prime }(0)=-2(f-u^{\ast },v)=0$ in the real case; varying real and imaginary parts gives the complex case. Therefore

The defining relation for $P_{\mathcal{U}}$ givesPart (a) therefore shows that $P_{\mathcal{U}}f$ is the best approximation. Moreover, $P_{\mathcal{U}}f$ and $f-P_{\mathcal{U}}f$ are orthogonal, soThus orthogonal projection is a contraction:

The integral form of the Taylor theorem about the left endpoint iswhere $q_{k-1}\in {\mathcal{P}}_{k-1}$. Apply the order-$k$ divided difference at $x_i,\dots ,x_{i+k}$. The polynomial term vanishes, and linearity permits interchange with the integral:By the definition $M_i(t)=k[x_i,\dots ,x_{i+k}](\cdot -t{)}_{+}^{k-1}$,This is the Peano kernel theorem for the divided-difference functional.

Substitute the B-spline expansion $s={\sum }_{j=1}^n{a}_j{M}_j$ into $(M_i,s)=k!{\gamma }_i$. The coefficients solvewith the Gram matrix and right-hand sideThe linearly independent B-splines have a positive-definite Gram matrix, so this system has a unique coefficient vector.

For any admissible $f$, part (a) rewrites the constraints asHence $f^{(k)}-s$ is orthogonal to every $M_i$ and therefore to their span, which contains $s$. The Pythagorean theorem in an inner-product space givesChoose any $k$-fold antiderivative $\sigma$ of $s$. The identity from part (a) and $(M_i,s)=k!{\gamma }_i$ show that $\sigma$ satisfies all prescribed divided differences, and equality holds in the norm bound. Thereforecharacterizes the minimizers. They are unique up to addition of an arbitrary polynomial in ${\mathcal{P}}_{k-1}$, which changes neither the order-$k$ divided differences nor the $k$th derivative.