While computing matrix elements of the evolution operator in Quantum Field Theory for the harmonic oscillator using the path integral formalism, I came across the assumption that all physically relevant paths are functions of a particular type, which I found puzzling. Therefore I would like to understand better what type of paths I am dealing with here. To be precise, let $t_1< t_2$ and $q_1,q_2$ be real numbers, then all paths of the following type are considered physical: paths given by functions (possibly by lack of a better word) $$ q(t) = f(t) + g(t), $$ where $f$ is a smooth real function obeying $f(t_1) = q_1$ and $f(t_2) = q_2$ and $g$ is of the form $$ g(t) = \sum_{k=1}^{\infty} a_k \sin\left(\pi k \frac{t-t_1}{t_2-t_1}\right). $$ The functions $\sin\left(\pi k \frac{t-t_1}{t_2-t_1}\right)$ form an orthonormal basis of $$ L^2_0=\left\{g\in L^2\left([t_1,t_2]\right)| g(t_1) = g(t_2) = 0\right\}, $$ but it seems obvious that not every set of coefficients $a_k$ will define a function $g\in L^2_0$: by choosing a strongly divergent series for the $a_k$ we will definitely create a series that does not converge for almost all $t$.
My question now is: what can we possibly say about the objects that are defined by this series when the $a_k$ are arbitrary real numbers? In other words, characterize the set given by $$ \left\{g(t) = \sum_{k=1}^{\infty} a_k \sin\left(\pi k \frac{t-t_1}{t_2-t_1}\right) | a_i \in \mathbb{R} \mbox{ for all } i \right\}. $$ I understand that this series does not converge for all possible sequences $a_k$: it might however in those cases be possible to interpret it as the formal series of some other object, such as a distribution. I am also interested in such interpretations.
I cannot say much more than that this set definitely encompasses $L^2_0\left([t_1,t_2]\right)$ but is most definitely bigger.