Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, and $\mathrm{Hol}_\beta(S^1)$ on the other hand (space of Hölder functions of some exponent $\beta<1$ with the usual Hölder norm).

Given a pair of functions $(f,g)\in L^2(\mu)\times \mathrm{Hol}_\beta(S^1)$, consider the following property: $$(*) \qquad \exists (\varphi_n) \mbox{ a sequence of smooth functions on the circle such that }\\ \varphi_n' \to f \mbox{ in } L^2(\mu) \quad\mbox{and}\quad \varphi_n\to g \mbox{ in } \mathrm{Hol}_\beta(S^1).$$ Edit: the first convergence is about the derivative of $\varphi_n$, the ' was forgotten in the first version of the question.

My question is about the following general question:

Which pairs $(f,g)$ satisfy property $(*)$?

Under the above assumptions ($\mu$ singular and $\beta<1$), I think that all pairs satisfy $(*)$, but the way I expect to be able to prove it would be somewhat tedious. It happens that I need such a result for some pairs, e.g. restricting to $g$ constant would be enough for my purpose.

My question: do you know a reference for this kind of result, or a way to prove such result without cutting in four too many epsilons, either in the general case or for some particular pairs?

Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, and $\mathrm{Hol}_\beta(S^1)$ on the other hand (space of Hölder functions of some exponent $\beta<1$ with the usual Hölder norm).

Given a pair of functions $(f,g)\in L^2(\mu)\times \mathrm{Hol}_\beta(S^1)$, consider the following property: $$(*) \qquad \exists (\varphi_n) \mbox{ a sequence of smooth functions on the circle such that }\\ \varphi_n' \to f \mbox{ in } L^2(\mu) \quad\mbox{and}\quad \varphi_n\to g \mbox{ in } \mathrm{Hol}_\beta(S^1).$$ Edit: the first convergence is about the derivative of $\varphi_n$, the ' was forgotten in the first version of the question.

My question is about the following general question:

Which pairs $(f,g)$ satisfy property $(*)$?

Under the above assumptions ($\mu$ singular and $\beta<1$), I think that many pairs satisfy $(*)$ (e.g. all pairs $(f,g)$ where $g$ is $\alpha$-Hölder for some $\alpha>\beta$) but the way I expect to be able to prove it would be somewhat tedious. It happens that I need such a result for some pairs, e.g. restricting to $g$ constant would be enough for my purpose.

My question: do you know a reference for this kind of result, or a way to prove such result without cutting in four too many epsilons, either in the general case or for some particular pairs?