Fourier multipliers and transfer operators (Q2031480)

The author gives a rigorous proof of a conjectured numerical value proposed by \textit{X. Chen} and \textit{H. Volkmer} [J. Fractal Geom. 5, No. 4, 351--386 (2018; Zbl 1400.37026)] which estimates a quantity related to the spectral radius of a transfer operator. The problem is significantly connected to the theory of Fourier multipliers. More specifically, the author takes the bounded linear operator \(\mathcal{L}: C^{0}([0, 1])\rightarrow C^{0}([0, 1])\) defined by \[(\mathcal{L}u)(t) = \frac{1}{3} \sum_{i=0}^{3}\left|\sin\left(\frac{\pi (t+i)}{3}\right)\right|u\left(\frac{t+i}{3}\right).\] For estimating the conjectured numerical value \(c=\lim_{n\rightarrow +\infty}||\mathcal{L}^{n}||^{1/n}\), the following complex function is used: \[d(z)=\exp\bigg(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\frac{1}{3^{n}-1}\sum_{j=0}^{3^{n}-1}\prod_{k=0}^{n-1}\sin\bigg(\frac{3^{k}j\pi}{3^{n}-1}\bigg)\bigg), \quad z\in \mathbb{C}.\] Note that \(d(z)\) extends analytically to \(\mathbb{C}\). The smallest positive zero \(\alpha>0\) is the reciprocal of the spectral radius \(c\), i.e., \(c=1/\alpha\). He describes a rigorous computation to determine a better estimate of \(c\), namely \[c= 0.648314752798325682324771447 \dots \pm 10^{-27}.\] The author also considers a more general form of the above bounded linear operator \(\mathcal{L}\) and estimates its spectral radius. He gives two applications to justify the importance of his results.