Is there a complete orthornomal basis of a Hilbert space which takes positive values on a discrete set?

Question

Is there a complete orthonormal basis $\{f_n\}$ (of continuous functions) of the Hilbert space of square integrable functions on $[0,\,\infty)$ for which there exists a countable set $S\subset [0,\,\infty)$ such that $\forall x \in S$ we have $f_n(x)\geq 0,\, \forall n?$

Or could anyone point me to a paper on a similar topic?

Thank you in advance.

Answer 1

Since two functions are equal, up to a set of measure zero, one can always do this. Take any orthonormal basis and a discrete set. Assign a positive value to the basis functions at every point in the discrete set. We are through now. Am I making it correctly? Correct me if it is wrong.