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

Question

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!

Answer 1

In general it is not true. Let $\{f_n\}_{n\geq 1}=\{1,z,\overline{z},z^2,\overline{z^2},\ldots\}$, then as the OP pointed out $a_n=o(n^{-k})$. However, with a suitable permutation $\sigma$ of the basis $\{f_n\}_{n\geq 1}$, we will have that coefficients in this new basis satisfy $\tilde{a}_n=a_{\sigma(n)}$. We can choose $\sigma$ so that for infinitely many $n$, $\sigma(n)\gg n$.Then it might happen for such $n$ that $$ \frac{\tilde{a}_n}{n^k}=\frac{a_{\sigma(n)}}{\sigma(n)^k}\left(\frac{\sigma(n)}{n}\right)^k\to \infty. $$ Indeed, although $a_{\sigma(n)}/\sigma(n)^k$ is small, $(\sigma(n)/n)^k$ might be very large. Providing a more explicit example from this sketch is now a simple exercise.

Answer 2

Here is a second example where a given basis fails to do the job. If $\mu$ has, say, a continuous positive density, then there a homeomorphism $h$ that sends $(\mathbb S^1,\mu)$ to $(\mathbb S^1,\nu)$ (uniform), in the sense $h^*\nu=\mu$. Now because of the result you cite, the usual basis $(\phi_n)$ in $(\mathbb S^1,\nu)$ detects $\mathcal C^k$ functions according to your criterion, hence its preimage $(\phi_n\circ h)$ in $(\mathbb S^1,\mu)$ detects functions $f$ such that $f\circ h$ is $\mathcal C^k$. If $h$ is not $\mathcal C^k$ (i.e. the density of $\mu$ is not $\mathcal C^{k-1}$, I suspect), then you will have functions satisfying your criterion but are not smooth.

If the density $\rho$ such that $\mathrm d\mu=\rho\mathrm d\nu$ is bounded above and below, then I believe $(I_n\cdot\phi_n/\rho)$ will do the trick, with $1/I_n=\|\phi_n/\rho\|^2$.