I would like to understand whether $$ \left\lbrace \sin (n\pi x) \, e^{-\sqrt{a-(n\pi)^2}x}\right\rbrace $$ for $a>0$, is an orthogonal bases for $L^2(0,1)$. Do you have any smart idea how to proceed, or maybe is there any useful known result?
2026-03-27 13:19:55.1774617595
Orthogonal bases for $L^2(0,1)$
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Sorry, I just realized it is clear it cannot be... just by considering the inner product between two of them with different index $n$...