Interpolation constraint in Hilbert space

Question

The paper "Geodesic Interpolating Splines" made the following comment on an interpolation problem:

Interpolation problem:

Let $\mathcal{H}$ be a Hilbert space, let $f_1, \dots, f_N \in \mathcal{H}$, and $c_1,\dots,c_N \in \mathbb{R}$ be given.

Find $h \in \mathcal{H}$ such that $\|h\|$ is minimum subject to the constraints $\langle f_i, h \rangle = c_i$ for $i=1,\dots,N$.

Comment:

It is indeed clear that the constraints are not affected if $h$ is replaced by $h + v$ where $v$ is orthogonal to all the $f_i$, so that the solution much be in fact be searched in the linear space spanned by $f_1, \dots, f_N$ and express the unknown $h$ as a linear combination $h = \sum_{i=1}^{N} \alpha_i f_i$.

I understand that the interpolation constraint will remain the same if we replace $h$ by $h + v$. But, how this property implies that the solution should be searched in linear space?

Is it possible to prove their claim with the elementary level knowledge of Hilbert space?

Answer 1

Let $P$ be the orthogonal projection onto the span of $f_i$. If $h$ satisfies the constraints, then so does $P(h)$. But, $||P(h)||\leq||h||$ with equality iff $h$ is in the span of the $f_i$. Thus, for any $h$ that satisfies the constraints outside the span, we can find an element of our Hilbert space with the smaller norm inside the span of the $f_i$.

Answer 2

Let $U$ denote the span of the elements $f_1,\dots,f_N$. We know that if $h \in H$ is a solution, then the element $h + v$ must also be a solution for any $v \in V$.

However, the solution $h$ can necessarily be written in the form $h = h_1 + h_2$ with $h_1 \in U$ and $h_2 \in U^\perp$. If we take $v = -h_2$, then we see that $h_1 = h + v$ must be a solution to the interpolation problem. In other words: if the interpolation problem has a solution $h$, then it must have a solution $h_1 \in U$.

Answer 3

If you can find minimizer $h$ outside of the $\text{span}\{f_1,\cdots,f_N\}$, then since $\langle f_i,h\rangle=0=c_i$, we have $c_1=\cdots=c_N=0$, which may be contradiction (if there exists $c_i\neq0$ given) or such $h$ does not exists (because if $\text{span}\{f_1,\cdots,f_N\}^\perp\neq\emptyset$, then take $a\in\text{span}\{f_1,\cdots,f_N\}^\perp$, and $||a/n||\rightarrow 0\in\text{span}\{f_1,\cdots,f_N\}$ satisfies the given conditions for all $n\in\mathbb{N}$), which is again a contradiction.