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Let $S_n^k$ denote de linear space of natural splines of degree $k$ and uniform grid $a < t_1 < \ldots < t_n < b$ over the bounded interval $[a,b] \subseteq \mathbb R$.

For any $m \in \mathbb N$ and $y \in \mathbb{R}^n$ there exists a unique $\sigma \in S^{2m-1}_n$ such that $\sigma(t_i) = y_i$. Such mapping is linear in $y$ and there exists a real symmetric positive semi definite matrix $\Omega$ such that $$\int_{a}^b (\sigma^{(m)}(t))^2\, dt = \langle y, \Omega y \rangle_{\mathbb R^n}.$$

In Utreras, F. (1983). Natural spline functions, their associated eigenvalue problem. Numerische Mathematik, they show that if $0 \leq \lambda_1 \leq \ldots \leq \lambda_n$ are the the eigenvalues of $\Omega$ then there exist positive constants $c_1$ and $c_2$ independent of $n$ such that

$$ \frac{1}{n} c_1 i^{2m} \leq \lambda_{i+m} \leq \frac{1}{n}c_2i^{2m} \quad i = 1, \ldots, n-m .$$

This estimates are important for obtaining asymptotics properties in nonparametric regression.

For $T(\sigma) = \sigma^{(m)}$ I suspect the the same statement holds for the eigenvalues of $T^*T$ where

$$ \| \sigma^{(m)} \|^2 = \langle \sigma, T^*T \sigma \rangle_S.$$

Does this statment or a similar version hold? It would be very useful to obtain a citable reference in such case.


My problem is that spline literature is huge and diverse and I cannot find proper information about the linear mapping $y \overset{f}{\to} \sigma$ which affects the eigenvalues of $T^*T$ since $\Omega = f^*T^*Tf$ and $$\|T(\sigma)\|^2 = \langle \sigma, (f^{-1})^*\Omega f^{-1}\rangle $$

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