Let \((s_n)\) be a sequence of real numbers. We say that \((s_n)\) is summable to \(\xi \) by the logarithmic power series method if$$\lim _{x\rightarrow 1^-}f(x)=\xi, \quad {\text{where}}\quad f(x)=-\frac{1}{\log (1-x)}\sum_{n=0}^{\infty }\frac{s_n}{n+1}x^{n+1}. $$It is well known that if the limit \(\lim_{n \rightarrow \infty }s_n=\xi \) exists, then the limit$$\lim _{x \rightarrow 1^-} f(x)=\xi $$also exists. In this paper, we determine Tauberian conditions of slowly decreasing type to obtain ordinary convergence of \((s_n)\) from its summability by logarithmic power series method. As a consequence of our result, we give a short proof of an earlier Tauberian theorem due to Kwee (Can J Math 20:1324–1331, 1968).

中文翻译：

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对数幂级数方法的慢减型陶伯条件

令\（（s_n）\）为实数序列。我们说，如果$$ \ lim _ {x \ rightarrow 1 ^-} f（x）= \ xi，\ quad {，通过对数幂级数方法，\（（s_n）\）可累加为\（\ xi \）。\ text {where}} \ quad f（x）=-\ frac {1} {\ log（1-x）} \ sum_ {n = 0} ^ {\ infty} \ frac {s_n} {n + 1} x ^ {n + 1}。$$众所周知，如果存在极限\（\ lim_ {n \ rightarrow \ infty s_n = \ xi \），则极限$$ \ lim _ {x \ rightarrow 1 ^-} f（x）= \ xi $$也存在。在本文中，我们确定缓慢减小类型的Tauberian条件以获得\（（s_n）\）的普通收敛性通过对数幂级数法从其可积性 由于我们的结果，我们简短地证明了由于Kwee而产生的更早的陶伯定理（Can J Math 20：1324–1331，1968）。