I'm interested in constructing a suitable interpolation operator that outputs smooth signals with "minimum energy". Let me clarify.
Let $s\ge1$. Let $x_{1},\dots,x_{n}$ be a (finite) collection of points in $[0,1]$. Given a corresponding collection of nodal values $y_{1},\dots,y_{n}$, we can define a map $f\in H^{s}(0,1)$ by requiring that
$$\|f\|_{H^{s}(0,1)}=\min\left\{\|g\|_{H^{s}(0,1)}\;|\;g\in H^{s}(0,1), \;\;g(x_{i})=y_{i}, \;\forall i=1,\dots,n\right\}.$$
This definition is well posed since the set $\left\{g\in H^{s}(0,1), \;\;g(x_{i})=y_{i}, \;\forall i=1,\dots,n\right\}$ is both convex and closed in $H^{s}(0,1)$, thus it admits a unique norm minimizer (minimum principle for Hilbert spaces).
Then, the "interpolation operator" under study is $Q:[y_{1},\dots, y_{n}]\mapsto f$, mapping $\mathbb{R}^{n}$ onto $H^{s}(\Omega)$. My question is about the properties of $Q$. In particular, is it linear?
My attempts so far. I've managed to show that $Q$ is homogeneous. In fact, let $\alpha\in\mathbb{R}$, $\mathbf{y}\in\mathbb{R}^{n}$. For simplicity, assume $\alpha\neq0$, $\mathbf{y}\neq0$ (otherwise the statement is trivial). Let $\|\cdot\|:=\|\cdot\|_{H^{s}(0,1)}$. By definition
$$\|\alpha Q(\mathbf{y})\|\ge \|Q(\alpha\mathbf{y})\|$$
since $\alpha Q(\mathbf{y})$ interpolates through the values $\alpha\mathbf{y}$. However, at the same time,
$$\|\alpha^{-1}Q(\alpha\mathbf{y})\|\ge \|Q(\mathbf{y})\|$$
since $\alpha^{-1}Q(\alpha\mathbf{y})$ passes through $(x_{i},y_{i})_{i=1}^{n}$. Putting the two together yields $\|\alpha Q(\mathbf{y})\|=\|Q(\alpha\mathbf{y})\|$. Then, by uniqueness, $\alpha Q(\mathbf{y})=Q(\alpha\mathbf{y})$, as wished.
However, I'm not able to prove that $Q(\mathbf{y}_{1}+\mathbf{y}_{2})=Q(\mathbf{y}_{1})+Q(\mathbf{y}_{2})$, and I'm starting to think that $Q$ might not be linear at all.
Additional comment. Since $H^{s}(0,1)$ is a reproducing kernel Hilbert space (RKHS), I'm wondering whether some "representer theorem" can be of help. The latter, in fact, show that for a given $\lambda>0$, the minimizer of
$$\|g\|+\lambda^{-1}\sum_{i=1}^{n}(g(x_{i})-y_{i})^{2}$$
exists unique, and it is given by $f_{\lambda}=\sum_{i=1}^{n}(\sum_{j=1}^{n}c_{i,j}y_{j})\phi_{i}$, for some functions $\{\phi_{i}\}_{i=1}^{n}\subset H^{s}(0,1)$, where $c_{i,j}$ are the entries of the matrix $C\in\mathbb{R}^{n\times n}$ given by $$C=(A^{T}A+\lambda A)^{-1}A,$$ where $A\in\mathbb{R}^{n\times n}$ is some fixed matrix (independent of $\lambda$ and $\mathbf{y}$).
Now the problem is that, in general, $f_{\lambda}$ does not interpolate the points. Furthermore, while we could use some "Lagrange-multiplier argument" to observe that there exists at least for some $\lambda_{*}>0$ for which $f_{\lambda_{*}}$ is an interpolant, and $Q(\mathbf{y})=f_{\lambda_{*}}$, this won't make $Q$ linear as, in general, $\lambda_{*}$ might depend on $\mathbf{y}$ (right?).
I have found that there exists a whole literature about interpolators with minimum norm. In particular, the following work by Boor and Lynch
"De Boor, C., & Lynch, R. E. (1966). On splines and their minimum properties. Journal of Mathematics and Mechanics, 15(6), 953-969."
shows, via the theory of Reproducing Kernel Hilbert Spaces (RKHS), that such interpolator is nothing but a projection $f\to Pf$ onto a subspace of suitable splines. The projection is done with respect to a suitable norm (different from $\|\cdot\|_{H^{s}}$) that ensures the interpolation constraints. Since, the function-to-grid-evaluation operator $E:f\to[f(x_{1}),\dots, f(x_{n})]$ is invertible over $S:=P(H^{s}(\Omega))$, we have $$Q=E^{-1}_{|S}E_{|S}Pf=E^{-1}_{|S}Ef.$$ In particular, $Q$ is linear, as conjectured.