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Suppose we have an orthogonal basis $\{e_k\}_{k \in \mathbb{N}}$ for $L^2[0,1]$. Do the functions \begin{align*} u_k(t) := \int_t^1e_k(s)\text{d} s,\quad k \in \mathbb{N}, \end{align*} also satisfy \begin{align*} \int_0^1u_i(s)u_j(s)\text{d} s = 0, \quad i \neq j? \end{align*} This is true for the the functions $\{\sqrt{2}\sin{(k - 1/2)\pi t}\}_{k \in \mathbb{N}}$ for example, and also seems to be true for the shifted Legendre polynomials. If it's not generally true, I'm curious under what conditions it might be true.

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Let $e_k(s) = e^{2 \pi i k s}$. Then the formula above gives $u_0(t)=1-t$, $u_1(t) = {1 \over 2 \pi i} (1-e^{2 \pi i s})$, and $\int_0^1 u_0(s) u_1(s) ds = -{1+i\pi \over 4 \pi^2}$.

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