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Assume we have a trigonometric series $$ f(x)=\sum_{n=1}^{\infty} a_n\sin nx \in \text{Lip }\alpha, \, 0<\alpha <1. $$ Is there anything we can say about the series $$ g(x)=\sum_{n=1}^{\infty} |a_n|\sin nx, $$ like $g\in $ Lip $\beta$ for some values of $0<\beta<1$ which depend on $\alpha$? For example, consider the following theorems that can be found in any trigonometric series monograph:

$\textbf{Theorem}$. Suppose that there is $0<\alpha <1$ such that $$ \sum_{k=n}^{\infty} |a_n|+|b_n| = O(n^{-\alpha}). $$ Then, the trigonometric series $$ h(x)=\sum_{n=0}^{\infty}a_n \sin nx + b_n \cos nx$$ belongs to the class Lip $\alpha$.

$\textbf{Theorem}.$ If the trigonometric series $h(x)$ belongs to the class Lip $\alpha$ with $1/2<\alpha<1$, then $$ \sum_{k=n}^{\infty} |a_n|+|b_n| = O(n^{-\alpha+1/2}). $$

If we combine both theorems in our setting, we obtain that if $f\in $ Lip $\alpha$, then $g\in $ Lip $\alpha-1/2$. Do you know any way to improve this so that we get something ''better'' than $\beta=\alpha-1/2$, or any reference to some paper? Thanks in advance.

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I would like to consider cosine series instead of sine series. If this very slight change of setting is permitted, then we can see quickly that a loss of $1/2$ in the Hölder exponent is possible, by putting together some facts about Fourier series.

First of all, there are functions $f\in\textrm{Lip}_{1/2}$ that are not in the Wiener algebra, that is, $\sum |b_n|=\infty$. (Once I have such a function, I can make it even, so it becomes a cosine series.) The Wikipedia article on convergence of Fourier series mentions this fact; a proof is given here, see Proposition 1.13.

For such an $f$, it is not possible to have $\sum |b_n|\cos nx \in\textrm{Lip}_{\alpha}$ for any $\alpha>0$ because this would imply pointwise convergence of the Fourier series (for example by Dini's test), and thus $b_n\in\ell^1$ by taking $x=0$.

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The trigonometric basis is not unconditional for the Lipschitz spaces. In fact, over the $L_p$ spaces, it is only unconditional for $p=2$. A similar result is true for their associated smoothness spaces, e.g. Lipschitz or Sobolev spaces.

A drop of $1/2$ in smoothness seems like the best you could hope for, given the embeddings between Lipschitz and $L_2$ Sobolev spaces.

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