Discrete Lebesgue constants (Q791081)

Discrete Lebesgue constants \(L_ m(q)\) with \(q\in {\mathbb{N}}\) are studied. An expression for them is \(L_ m(q)=(m+1)/q+\frac{1}{q}\sum^{q- 1}_{\ell =1}| \sin(\pi(m+1)\ell /q)| /\sin(\pi \ell /q).\) Two theorems and a corollary are proved. Theorem 1: For all \(q\geq 2m+2\) the estimates \(\frac{1}{\pi}\log(m+1)+0(1)\leq L_ m(q)\leq \frac{4}{\pi^ 2}\log(m+1)+0(1)\) hold, and they are the best estimates: \(L_ m(2m+2)=\frac{1}{\pi}\log(m+1)+0(1), L_ m((m+1)^ 2)=\frac{4}{\pi^ 2}\log(m+1)+0(1).\) Theorem 2: Let \(m_ n,q_ n\in {\mathbb{N}}\), \(m_ n\to \infty\), \(m_ n/q_ n\to \alpha \in [0,1/2]\). Then a) if \(\alpha\) is an irrational number or if \(\alpha =0\), then \[ \lim_{n\to \infty}(L_{m_ n}(q_ n)/\log(m_ n+1))=4/\pi^ 2; \] if \(\alpha =m/q\neq 0\), \((m,q)=1\), then \[ \frac{2}{\pi q}ctg\frac{\pi}{2q}\leq \lim \inf_{n\to \infty}[L_{m_ n}(q_ n)/\log(m_ n+1)]\leq \lim \sup_{n\to \infty}L_{m_ n}(q_ n)/\log(m_ n+1)\leq \frac{4}{\pi^ 2}, \] and b) for each \(\gamma\in [\frac{2}{\pi q}ctg\frac{\pi}{2q},\frac{4}{\Pi^ 2}]\) there exist m'\({}_ n,q'\!_ n\in {\mathbb{N}}\) that m'\({}_ n/q'\!_ n\to m/q\) and \(\lim_{n\to \infty}(L_{m'\!_ n}(q'\!_ n)/\log(m_ n\!'+1))=\gamma.\) Corollary: The equality \(\lim \inf_{m\to \infty,m/q\to \alpha}(L_ m(q)/\log(m+1))=C(\alpha)\) holds, where \(C(\alpha)=4/\pi^ 2\) if \(\alpha\) is irrational or if \(\alpha =0\) and \(C(\alpha)=\frac{2}{\pi p}ctg\frac{\pi}{2p}\) if \(\alpha =r/p\neq 0\), \((r,p)=1\) is a Riemann type function.