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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?

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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 $n$...

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