If in the Fourier expansion of a function, only the first terms are important and one can ignore higher terms (for example as below), what does it meana and what can one say about the function $f$(\theta)?
$f(\theta) \simeq a \sin(\theta) + b \sin(2\theta) + c \sin(3\theta)$
"Importance" depends on the application. For "well-behaved" functions $g$ (for example $g \in L^2$) the magnitude of the coefficients eventually has to drop, that is why you can in practical applications ignore the higher order terms, where the highest order to be considered depends on your application's required precision.
Having said that, it should be clear that the "lower order" terms, that is the terms not to be ignored, do not have any "special" interpretation that distinguishes them from the "ignored" terms, as the only criterion for the selection of the maximum order considered is the required precision.
The only exception here is the first coefficient, which represents the average value of the function $g$, but that is so in general and has nothing to do with which coefficients / terms are ignored.