Uniform convergence criterion for non-harmonic sine series,Sbornik: Mathematics

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Uniform convergence criterion for non-harmonic sine series
Sbornik: Mathematics ( IF 0.8 ) Pub Date : 2021-03-08 , DOI: 10.1070/sm9445
K. A. Oganesyan _{1,

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Affiliation

We show that for a nonnegative monotonic sequence $\{c_k\}$ the condition $c_kk\to 0$ is sufficient for the series $\sum_{k=1}^{\infty}c_k\sin k^{\alpha} x$ to converge uniformly on any bounded set for $\alpha\in (0,2)$ , and for any odd $\alpha$ it is sufficient for it to converge uniformly on the whole of $\mathbb{R}$ . Moreover, the latter assertion still holds if we replace $k^{\alpha}$ by any polynomial in odd powers with rational coefficients. On the other hand, in the case of even $\alpha$ it is necessary that $\sum_{k=1}^{\infty}c_k<\infty$ for the above series to converge at the point $\pi/2$ or at $2\pi/3$ . As a consequence, we obtain uniform convergence criteria. Furthermore, the results for natural numbers $\alpha$ remain true for sequences in the more general class $\mathrm{RBVS}$ .

Bibliography: 17 titles.

中文翻译：

非谐波正弦级数的统一收敛准则

我们证明，对于非负单调序列， $\ {C_K \}$ 该条件 $c_kk \ 0$ 足以使该级数 $\ sum_ {K = 1} ^ {\ infty} C_K \罪K ^ {\阿尔法}×$ 在的任何有界集上一致收敛 $\阿尔法\在（0,2）$ ，并且对于任何奇数， $\alpha$ 它足以在的整体上一致收敛 $\mathbb{R}$ 。此外，如果我们用 $K ^ {\阿尔法}$ 任何具有有理系数的奇次幂多项式替换，后一个断言仍然成立。在另一方面，在连的情况下 $\alpha$ 有必要使 $\ sum_ {K = 1} ^ {\ infty} C_K <\ infty$ 对上述一系列会聚在点 $\pi/2$ 或在 $2\pi/3$ 。因此，我们获得了统一的收敛标准。此外，对于 $\alpha$ 更一般的类中的序列，自然数的结果仍然正确 $\ mathrm {RBVS}$ 。

参考书目：17 个标题。

更新日期：2021-03-08

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