# Eigenvalues associated to differential opertator over natural spline spaces

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 $$\|T(\sigma)\|^2 = \langle \sigma, (f^{-1})^*\Omega f^{-1}\rangle$$