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Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to obtain $$ \sum_{k=1}^\infty |c_k| < 1 \ ? $$

As pointed out in the comments below, this is related to characterizing the functions in the unit ball of the Wiener algebra.

I am aware of the estimate $|c_k| \lesssim k^{-m}$ if $f \in C^m(0,1)$, but I'm looking for something more precise as above.

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    $\begingroup$ The sum looks like the norm of $f$ in the Wiener algebra, so I believe any space that embeds into the Wiener algebra with an explicit constant will give you a sufficient condition, and other than that not much can be said. $\endgroup$ Jan 24, 2021 at 21:13
  • $\begingroup$ @MateuszKwaśnicki I don't know exactly what you mean. $\endgroup$
    – Riku
    Jan 24, 2021 at 21:51
  • $\begingroup$ If $f$ is extended to an even $2$-periodic function, then the sum $\|f\|_{\mathbb A} = \sum_k |c_k|$ is the norm of $f$ in the Wiener algebra $\mathbb A$ (up to multiplication by a constant, perhaps). So the condition $\|f\|_{\mathbb A} < 1$ describes the unit ball in the Wiener algebra $\mathbb A$. Now we know, for example, that if $\alpha > \frac12$, then the space $C^\alpha$ of Hölder-continuous functions with exponent $\alpha$ is contained in $\mathbb A$ when $\alpha > 1$, with $\|f\|_{\mathbb A}$ bounded by a constant $C_\alpha$ times the Hölder norm $\|f\|_{C^\alpha}$. (1/2) $\endgroup$ Jan 24, 2021 at 22:09
  • $\begingroup$ Therefore, if $\|f\|_{C^\alpha} < 1 / C_\alpha$, then of course $\|f\|_{\mathbb A} < 1$. If we knew the value of $C_\alpha$, this would give a sufficient condition you are looking for. The constant can be found by inspecting the proof of Bernstein's theorem mentioned above. See, for example, Zygmund's Trigonometric series (one of the online sources I found mentioned it is on page 241 of the 1988 edition). (2/2) $\endgroup$ Jan 24, 2021 at 22:15
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    $\begingroup$ "Laplacian with Neuman boundary conditions" is a fancy way to say that $\phi_n(x)=\cos\pi n x$? $\endgroup$ Jan 25, 2021 at 0:39

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