White noise gain of uniform spline interpolation
This figure is the squared norm or energy of the impulse response:

$$
\|h(t)\|^2=\int_{\mathrm{R}} |h(t)|^2 dt
$$

As you point out correctly, this is the noise amplification. Considering a signal

$$
r(t) = x(t) + n(t)
$$
with $x(t)$ is the signal of interest and $n(t)$ is AWGN with variance $\sigma^2$. Then, after filtering

$$
\begin{align}
y(t) &amp;= h(t)*r(t)\\
&amp;=h(t)*x(t) + \int_{\mathrm{R}} h(t-\tau)n(\tau)d\tau\\
&amp;=h(t)*x(t) + \tilde{n}(t)
\end{align}
$$

with $\tilde{n}(t)$ being the filtered noise. Now, considering

$$
\begin{align}
E[\tilde{n}(t)\tilde{n}^*(t)] &amp;= \int_\mathrm{R}\int_\mathrm{R} h(t-\tau_1)h^*(t-\tau_2)E[n(\tau_1)n(\tau_2)]d\tau_1 d\tau_2\\
&amp;=\sigma^2\int_{\mathrm{R}^2}h(t-\tau_1)h^*(t-\tau_2)\delta(\tau_1-\tau_2)d\tau_1d\tau_2\\
&amp;=\sigma^2\int_\mathrm{R}h(t-\tau_1)h^*(t-\tau_1)d\tau_1\\
&amp;=\sigma^2\|h(t)\|^2
\end{align}
$$

where we have used that the noise is white, i.e. its autocorrelation is a delta-function. Note that after filtering generally the noise is not white anymore, but colored.