Question

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 \[ \left(f(x)+\cos^{3}(x)\right)^{2}=1. \]

Answer 1

Let $\|\cdot \|_F $ denote the $\ell^1$ norm of the Fourier series of a function on $[0,2\pi]$. This is a Banach algebra norm on functions whose Fourier series converge absolutely. Note that $$1 + \cos^2(x) + \cos^4(x) = \frac{15}{8} + \frac{1}{2} e^{2ix} + \frac{1}{2} e^{-2ix} + \frac{1}{16} e^{4ix} + \frac{1}{16} e^{-4ix}$$ Write this as $(15/8)(1 + W(x))$, where $\|W(x)\|_F = 3/5$. So $$g(x) = 3 (15/8)^{-1/2} \cos^5(x) (1+W(x))^{-1/2}= 3 (15/8)^{-1/2} \cos^5(x)\sum_{k=0}^\infty {{-1/2} \choose k} W(x)^k$$ Thus $$\|g\|_F \le 3 (15/8)^{-1/2} \sum_{k=0}^\infty \left| {{-1/2} \choose k}\right| (3/5)^k = 2 \sqrt{3}$$ Moreover, by explicitly evaluating the norm of a partial sum of the series and bounding the norm of the remainder you can get arbitrarily good approximations to the actual value of $\|g\|_F$.