I am inquiring about a particular result shown in section 2.1 of Grace Wahba's book (pages 23-24), Spline models for observational data.
The setup is as follows. We consider the data model
$$y_i = f\left(\frac i n \right) + \epsilon_i, \quad \text{for } i=1,\ldots, n.$$ We assume $f$ belongs to a Sobolev space with periodic boundary conditions. The author seeks a solution to this, in the form
$$f_\lambda(t) = a_0 + \sum_{k=1}^{n/2 -1} \left( a_k \sqrt{2} \cos(2 \pi kt) + b_k\sqrt{2} \sin (2 \pi k t) \right) + a_{n/2} \cos (\pi n t)$$
minimizing the following penalized criterion
$$\frac 1n \sum_{i=1}^n \left(y_i - f \left( \frac i n \right)\right)^2 + \lambda \int_0^1 (f^{(m)} (u))^2 du$$ for $\lambda >0$ and a nonnegative integer $m$.
The author then derives optimal coefficient values. My question pertains to only two of them, so I will ignore the rest. Let $k = 1, \ldots, n/2 - 1$. Then:
$$a_k = \frac{\sqrt{2}}{n} \sum_{i=1}^n \cos\left(2 \pi k \frac i n\right) f \left( \frac i n\right)$$ $$b_k = \frac{\sqrt{2}}{n} \sum_{i=1}^n \sin\left(2 \pi k \frac i n\right) f \left( \frac i n\right).$$
Next the author then defines the estimates of these terms, given by replacing the $f$ term with the data:
$$\hat{a}_k = \frac{\sqrt{2}}{n} \sum_{i=1}^n \cos\left(2 \pi k \frac i n\right)y_i$$ $$\hat{b}_k = \frac{\sqrt{2}}{n} \sum_{i=1}^n \sin\left(2 \pi k \frac i n\right) y_i.$$
The author redefines the minmization criterion (in a step I don't fully understand myself), and then claims the minimizing values are of the form
$$a_k = \hat{a}_k / (1 + \lambda (2 \pi k)^{2m})$$ $$b_k = \hat{b}_k / (1 + \lambda (2 \pi k)^{2m})$$
Then, the estimated function with penalty $\lambda$ (ignoring the other terms, and written exactly as it appears in the text otherwise):
$$f_\lambda (t) \propto \sum_{k=1}^{n/2 -1}\left( \frac{\hat{a}_k}{1 + \lambda (2 \pi k)^{2m}} \cos(2 \pi kt) + \frac{\hat{b}_k}{1 + \lambda (2 \pi k)^{2m}}\sqrt{2} \sin (2 \pi k t) \right) .$$
Considering this is a highly cited text with seemingly no errata anywhere, I wanted to ask: is there a factor of $\sqrt{2}$ missing in the final expression for $f_{\lambda}(t)$? Such a factor appears in the solution form definition, but disappears in the final expression.
I implemented this procedure in R and found that the extra $\sqrt{2}$ which appears missing is necessary for a good fit, and without it, the amplitude does not match the data.
