Analogue of decay of Fourier coefficients of a smooth function on $\mathbb{S}^1$.

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!

• 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. Jul 25, 2020 at 7:10