Approximation of the Lebesgue constant of the Fourier operator by a logarithmic-fractional-rational function (Q6549001)

Let \N\[\NL_n=\frac{2}{\pi}\int_{0}^{\pi/2}\frac{|\sin (2n+1)t|}{\sin t}dt,\quad n=0,1,\ldots. \N\]\NThe author considers the approximation of the Lebesgue constants \(L_n\) by a logarithmic-fractional-rational function of the form \N\[\Ny_n(a,b,c)=\frac{4}{\pi^2} \log (n+a)+b+ \frac{c}{(n+a)^2} \N\]\Nwhere \((a,b,c)\in Y\) and \(Y=[0,1]\times [0, 1.5]\times [0, 0.1]\). The approximate equality \N\[\NL_n\approx \frac{4}{\pi^2} \log (n+0.5)+\alpha+ \frac{\beta}{(n+0.5)^2}=y_n^{\ast} \N\]\Nis studied with \(\alpha=1.270353244\ldots\), \(\beta=0.002945386\ldots\). It is shown, in particular, that \N\[\NL_n> \frac{4}{\pi^2} \log (n+0.5)+\alpha+ \frac{\beta}{(n+0.5)^2}\quad\text{for}\quad n\geq 2, \N\]\N\[\N\lim_{n\to \infty} (L_n-y_n^{\ast})=0,\quad \sup_{n\in \mathbb{N}} (L_n-y_n^{\ast})=0.000005282\ldots, \N\]\Nand \N\[\N\inf_{(a,b,c)\in Y} \sup_{n\in \mathbb{N}} |L_n-y_n(a,b,c)|<0.000005283. \N\]\NThe results obtained substantially sharpens the results of [\textit{I. A. Shakirov}, Russ. Math. 66, No. 5, 70--76 (2022; Zbl 1524.41075); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2022, No. 5, 86--93 (2022); \textit{D. Zhao}, J. Math. Anal. Appl. 349, No. 1, 68--73 (2009; Zbl 1154.26027); \textit{C.-P. Chen} and \textit{J. Choi}, Appl. Math. Comput. 248, 610--624 (2014; Zbl 1338.41020)].