Title: On some Féjer-type trigonometric sums Show Chinese title

Title: 关于某些Féjer型三角级数

Abstract: We examine the four F\'ejer-type trigonometric sums of the form \[S_n(x)=\sum_{k=1}^n \frac{f(g(kx))}{k}\qquad (0<x<\pi)\] where $f(x)$, $g(x)$ are chosen to be either $\sin x$ or $\cos x$. The analysis of the sums with $f(x)=g(x)=\cos x$, $f(x)=\cos x$, $g(x)=\sin x$ and $f(x)=\sin x$, $g(x)=\cos x$ is reasonably straightforward. It is shown that these sums exhibit unbounded growth as $n\to\infty$ and also present `spikes' in their graphs at certain $x$ values for which we give an explanation. The main effort is devoted to the case $f(x)=g(x)=\sin x$, where we present arguments that strongly support the conjecture made by H. Alzer that $S_n(x)>0$ in $0<x<\pi$. The graph of the sum in this case presents a jump in the neighbourhood of $x=2\pi/3$. This jump is explained and is quantitatively estimated when $n\to\infty$. Show Chinese abstract

Abstract: 我们研究了四种形式为 \[S_n(x)=\sum_{k=1}^n \frac{f(g(kx))}{k}\qquad (0<x<\pi)\] 的 Féjer 型三角和，其中 $f(x)$, $g(x)$ 被选取为要么是 $\sin x$ 要么是 $\cos x$。 对包含$f(x)=g(x)=\cos x$、$f(x)=\cos x$、$g(x)=\sin x$和$f(x)=\sin x$、$g(x)=\cos x$的和的分析相当直接。文中表明，当$n\to\infty$增大时，这些和表现出无界增长，并且它们的图形在某些$x$值处出现“尖峰”，我们对此给出了解释。 主要的工作集中在情况$f(x)=g(x)=\sin x$，我们提出了强有力的支持 H. Alzer 提出的猜想的论据，即$S_n(x)>0$在$0<x<\pi$中成立。 在这种情况下，和的图形在$x=2\pi/3$附近出现跳跃。 当$n\to\infty$时，对该跳跃进行了解释，并进行了定量估计。