Let $$ f(x)=\sin(x)\sqrt{1+\cos^{2}(x)+\cos^{4}(x)}. $$ In my study of almost commuting unitary matrices, $U$ and $V$, I have need for a bound like $$ \left\Vert \tilde{f}(V)U-U\tilde{f}(V)\right\Vert \leq C\left\Vert VU-UV\right\Vert , $$ where $\tilde{f}(e^{\pi ix})=f(x)$ so my functional calculus makes sense. (Matrix function to any applied listeners). The norm is the operator norm.

This is all in the context spin Chern numbers and the Pfaffian-Bott index, as in my paper with Hastings, Topological insulators and $C^{*}$- algebras: Theory and numerical practice'' which came out in 2011. I want precise bounds on how small a commutator I need to ensure a derived matrix remains invertible.

One way to estimate $C$ is to writing $f$ and $f^{\prime}$ as Fourier series and working term by term. Let $g=f^{\prime},$ so $$ g(x)=\frac{3\cos^{5}(x)}{\sqrt{1+\cos^{2}(x)+\cos^{4}(x)}}. $$ I need to know, or get a good estimate on $\left\Vert \hat{g}\right\Vert _{1}$, meaning the $\ell^{1}$-norm of the fourier series of $g(x)$.

What is the $\ell^{1}$-norm of the Fourier series of $\frac{3\cos^{5}(x)}{\sqrt{1+\cos^{2}(x)+\cos^{4}(x)}}$, or what is a good upper bound?

I can prove this is less than $3\pi$ but I want a tighter estimate. Perhaps someone has seen this before. Is there a stategy to bound this I should follow? Better still, what is the Fourier series for $g$?

Any hints on finding $C$ by another route will be welcome.

A little more about $f$: The Bott index and its relatives require three functions with $f^2 + g^2 + h^2 = 1$ and $gh=0$ and these are to be continuous, real valued and periodic. And non-trivial, so $h=0$ is not allowed. For theoretical work these are all equally valid, but to use these in index studies of finite systems we need to pick carefully to control a bunch of commutators. This $f$ is designed to be similar to $\sin$ but will big "flat spots." Notice $$ \left(f(x)+\cos^{3}(x)\right)^{2}=1. $$