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minimum-energy interpolation
In bivariate spline approximation, minimum-energy spline interpolation is of
some interest (see, e.g., FasshauerSchumaker96a).
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In the univariate context, this amounts to choosing, from some subspace $S$
of the continuous piecewise polynomials with break sequence $t:= (t_i: i)$, the
element $f$ that matches a given $g$ at the $t_i$ and otherwise minimizes the
seminorm whose square is $\sum_i \int_{t_i}^{t_{i+1}} D^2f$. Here, it is assumed
that the map $S\to \RR^t f\mapsto (f(t_i): i)$ is onto. Hence, in particular,
the order, $k$, of the spline space is at least 2.
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If $S$ consists of $\Pi_{k,t}^0$, the space of all pp's of order $k$ with
break sequence $t$, then the minimum-energy interpolant is the broken-line
interpolant, regardless of $k>1$. In particular, the order of approximation
is only 2, regardless of the smoothness of $g$.
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If $S$ consists of $\Pi_{k,t}^\rho$, the space of all pp's of order $k$ with
$\rho$ continuous derivatives, for some $\rho>0$ and some $k>3$, then $S$ is
a subspace of $L_2^{(2)}$ and, correspondingly, the minimum-energy
interpolant is just the `natural' cubic spline interpolant. Hence, the order
of approximation is again only 2 (though it would be 4 away from the
endpoints). This leaves the case $\rho=1$, $k=3$, as an interesting
intermediate one. %
There is work on the related scheme in which one minimizes the 1-norm, in
particular, the 1-norm of $Df$.
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