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Let $\nu$ be the uniform measure on the unit circle $\mathbb{S}^1 \subset \mathbb{R}^2$, normalised so that $\nu(\mathbb{S}^1) = 1$. Suppose $\mu$ is a Borel probability measure on $\mathbb{S}^1$ which is absolutely continuous w.r.t. $\nu$, that is $\mu \ll \nu$. Let $\{f_n\}_{n\geq 1}$ be an orthonormal basis for $L^2(\mathbb{S}^1,\mu)$. Is it true that for $g \in C^k(\mathbb{S}^1)$ $$ \int_{[0,2\pi]} f_n(\theta) g(\theta)d\mu(\theta) = o(1/n^k). $$ Or is it possible to choose an ONB such that the above holds? My question is motivated by the case when $\mu = \nu$ and the ONB is $\{1,z,\overline{z},z^2,\overline{z^2},\ldots\}$, where it is known to be true (See this.)

Thanks!

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    $\begingroup$ I think the type of coefficient decay you're describing for $C^k$ functions is unlikely to hold unless additional conditions are put on the measure $\mu$ at least. Coefficient decay with respect to the basis $1, z, \overline{z}, \ldots$ for $L^2(\nu)$ follows from integration by parts. So unless both the measure $\mu$ and the ONB are suitably "nice" with respect to the differentiable structure of $\mathbb{S}^1$, I wouldn't expect anything in particular for the coefficients of $C^k$ functions. $\endgroup$
    – Jason
    Jul 25, 2020 at 7:10

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