论文标题
在某些féjer型三角学金额上
On some Féjer-type trigonometric sums
论文作者
论文摘要
我们检查表格\ [s_n（x）= \ sum_ {k = 1}^n \ frac {f（g（kx））} {k} {k} {k} \ qquad（0 <x<π）\ e $ f（x）$ cos $ cos $ cos $ cose x的四个féjer-type三角学总和。用$ f（x）= g（x）= \ cos x $，$ f（x）= \ cos x $，$ g（x）= \ sin x $和$ f（x）= \ sin x $，$ g（x）= \ cos x $分析总和是合理的。结果表明，这些总和以$ n \ to \ infty $的形式表现出无限的增长，并且在其图表中以某些$ x $值表示“尖峰”，我们给出了解释。主要的努力专门用于$ f（x）= g（x）= \ sin x $的情况，我们提出了强烈支持H. alzer的猜想的参数，即$ s_n（x）> 0 $ in $ 0 <x <x <π$。在这种情况下，总和的图显示了$ x =2π/3 $的附近的跳跃。解释了此跳跃，并在$ n \ to \ infty $时进行定量估算。
We examine the four Féjer-type trigonometric sums of the form \[S_n(x)=\sum_{k=1}^n \frac{f(g(kx))}{k}\qquad (0<x<π)\] 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 $0<x<π$. The graph of the sum in this case presents a jump in the neighbourhood of $x=2π/3$. This jump is explained and is quantitatively estimated when $n\to\infty$.
