Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. Suppose that $\{a_k\}_{k=1}^{\infty} \subset \mathbb R$ satisfies $\sum_{k=1}^{\infty} |a_k|^2 <\infty$.
Does there exist a function $f\in L^2((0,1))$ with $\textrm{supp}\,f \subset (0,\frac{1}{2})$ such that$$ \int_0^1 f(x)\, \phi_{n^2}(x)\,dx = a_n \quad \text{for $n=1,2,\ldots$}?$$
