论文标题
非谐波正弦系列的均匀收敛标准
Uniform convergence criterion for non-harmonic sine series
论文作者
论文摘要
我们表明，对于非负单调序列$ \ {c_k \} $，条件$ c_kk \ to 0 $足以足以均匀地收敛$ \ sum_ {k = 1}^{\ infty} c_k \ sin k^αx $ for $ n $ n $ ach $ ach $α\ in oft and（and oft in onde），以及0,2）$α\ in（and）（0,2）整个$ \ mathbb {r} $上的均匀收敛。此外，如果我们用任何具有合理系数的奇数功率代替$ k^α$，则后一个断言仍然存在。另一方面，如果均为$α$，则有必要$ \ sum_ {k = 1}^{\ infty} c_k <\ infty $在$π/2 $或点$2π/3 $时在上述系列上收敛。因此，我们获得统一的收敛标准。此外，自然$α$的结果对于更通用的RBVS类序列仍然是正确的。
We show that for a nonnegative monotone sequence $\{c_k\}$ the condition $c_kk\to 0$ is sufficient for uniform convergence of the series $\sum_{k=1}^{\infty}c_k\sin k^α x$ on any bounded set for $α\in (0,2)$, and for an odd natural $α$ it is sufficient for uniform convergence on the whole $\mathbb{R}$. Moreover, the latter assertion still holds if we replace $k^α$ by any polynomial in odd powers with rational coefficients. On the other hand, in the case of an even $α$ it is necessary that $\sum_{k=1}^{\infty}c_k<\infty$ for convergence of the mentioned series at the point $π/2$ or at the point $2π/3$. Consequently, we obtain uniform convergence criteria. Besides, the results for a natural $α$ remain true for sequences from more general RBVS class.
