Can (some) linear regression model this (population) function accurately? James, Witten, Hastie, Tibshirani on page 35 of their book state with reference to the figure below:

In Figure 2.11, the true f [given by the black curve] is substantially
non-linear, so no matter how many training observations we are given,
it will not be possible to produce an accurate estimate using linear
regression.


Is this statement correct? Keeping in mind the widespread misunderstanding that linear regression can only model linear relationships/missing the fact that linear regression refers to linearity in parameters and can accommodate a large collection of non-linearities, I wonder whether this statement is correct. And if it is, I wonder what it is about this particular shape of f that puts it out of reach for the non-linearities that can be accommodated by linear regression?
page 35:

"... linear regression assumes that there is a linear relationship
between Y and $X_1$,$X_2$,...,$X_p$."

page 90:

"The linear regression model assumes that there is a straight-line
relationship between the predictors and the response."

On page 91 they call the model in 3.36 a linear model, even with an exclamation mark.
page 91:

(3.36) $mpg=\beta_0+\beta_1*horsepower+\beta_2*horsepower^2+\epsilon$
Equation 3.36 involves predicting mpg using a non-linear function of
horsepower. But it is still a linear model! That is, (3.36) is
simply a multiple linear regression model with $X_1=horsepower$ and
$X_2=horsepower^2$.


A straight line is always going to leave something to be desired. In that sense, the claim is correct.
However, the true relationship looks cubic, which means that $\mathbb E[Y\vert X] = \beta_0 + \beta_1X + \beta_2X^2 + \beta_3X^3$ might be a reasonable model. Since this is linear in the parameters, the model is a linear regression, yet the fit should be reasonable. In that sense, the claim in incorrect.
Once you get into function convergence theorems in mathematical analysis, you will see that linear combinations of polynomial terms can get arbitrarily close to an awful lot of functions.
