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.


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 $x=0$.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.