= Solution
For ordinary homoscedastic <fixed-design nonparametric regression>, take observations $Y_i=g(x_i)+\epsilon_i$ at known distinct sites, with independent mean-zero errors of common <variance>. A degree-$p$ <local polynomial estimator> at a target $u$ solves the finite-dimensional <weighted least squares> problem
$$
\widehat\beta(u)=\operatorname*{argmin}_{\beta\in\mathbb R^{p+1}}
\sum_i K\!\left(\frac{x_i-u}{h}\right)
\left[Y_i-\sum_{r=0}^p\beta_r(x_i-u)^r\right]^2.
$$
Here $K$ is a nonnegative <regression kernel>, $h>0$ is the <smoothing bandwidth>, and sufficient local design rank ensures uniqueness. The estimate of the <regression function> is $\widehat g(u)=\widehat\beta_0(u)$. Each target has its own locally fitted <polynomial>; this is not one global <polynomial> fit. The bandwidth controls the localization and the <bias-variance tradeoff>.
A <natural cubic smoothing spline> instead solves the global penalized problem
$$
\widehat g_\lambda=\operatorname*{argmin}_{v\in H^2[a,b]}
\left\{\sum_i[Y_i-v(x_i)]^2+\lambda\int_a^b[v''(x)]^2\,dx\right\},\qquad\lambda>0.
$$
The <Sobolev space> here consists of functions with square-integrable weak <derivatives> through order two. The minimizer is a <natural cubic spline> with knots at the observation sites and linear tails outside the extreme sites. Equivalently one may perform the minimization on that finite-dimensional spline space: the <minimum roughness property of the natural cubic spline interpolant> shows that replacing a candidate by the natural spline with the same site values preserves the data loss and decreases the second-<derivative> penalty. The <smoothing parameter> $\lambda$ controls curvature; it differs from the local bandwidth $h$.
For the interval-average observation problem, the printed upper limit in the loss is an indexing error: there are only $n-1$ responses and intervals, so a term with $i=n$ would involve the undefined $Y_n$ and $x_{n+1}$. The coherent loss uses the upper limit $n-1$. Put $\Delta_i=x_{i+1}-x_i$ and define the linear observation functionals
$$
L_iv=\frac1{\Delta_i}\int_{x_i}^{x_{i+1}}v(x)\,dx,\qquad1\leq i\leq n-1.
$$
The intended criterion on $C^1[a,b]$ is therefore
$$
S_\lambda(v)=\sum_{i=1}^{n-1}(Y_i-L_iv)^2+\lambda\int_a^b[v'(x)]^2\,dx.
$$
Unlike the ordinary <cubic smoothing spline>, this uses interval averages as data and a first-<derivative> penalty; the resulting spline degree will be two.
Fix any candidate $v\in C^1[a,b]$. By the supplied interpolation property, there is a unique <quadratic spline> $q\in C^1[a,b]$ with knots $x_1,\ldots,x_n$, constant on both exterior intervals, such that $L_iq=L_iv$ for every $i$. Put $d=v-q$. Then
$$
\int_{x_i}^{x_{i+1}}d(x)\,dx=0\qquad(1\leq i\leq n-1).
$$
On each interior knot interval $q''$ is a constant $c_i$, and on the two exterior intervals $q'=0$. Integrate $q'd'$ by parts separately on the intervals. At every knot the boundary contributions cancel because both $q'$ and $d$ are continuous; the contributions at $a$ and $b$ vanish because $q'$ is zero there. Thus
$$
\int_a^bq'(x)d'(x)\,dx
=-\sum_{i=1}^{n-1}c_i\int_{x_i}^{x_{i+1}}d(x)\,dx=0.
$$
This orthogonality is the key to the <quadratic smoothing spline for interval averages>. Expanding the roughness penalty gives
$$
\int_a^b[v'(x)]^2\,dx
=\int_a^b[q'(x)]^2\,dx+\int_a^b[d'(x)]^2\,dx.
$$
The data-fit terms are equal because the interval averages are equal. Consequently,
$$
\boxed{S_\lambda(v)=S_\lambda(q)+\lambda\int_a^b[d'(x)]^2\,dx\geq S_\lambda(q).}
$$
Equality can occur only if $d'=0$ everywhere: the <derivative> is continuous and its squared integral is zero. Then $d$ is constant, and any of its zero interval averages forces that constant to be zero. Therefore every minimizer must equal its matching <quadratic spline> $q$.
We conclude that \b[the minimizer is a continuously differentiable quadratic spline with knots at the specified sites and constant tails on both exterior intervals.] Existence and uniqueness also follow from the supplied interpolation property. The interval-average map identifies the finite-dimensional space of these <quadratic splines> with $\mathbb R^{n-1}$, and the criterion becomes a <strictly convex> quadratic function of that average vector: its data-fit term is $\|Y-y\|^2$, while its penalty is nonnegative and quadratic. It has a unique minimizer, and the strict equality argument above proves uniqueness in the full function class.
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