AI 中文总结
该研究对Zalcwasser1936年提出的两个问题给出肯定答案，证明了Fourier部分和沿特定多项式子序列的(C,α)平均在L¹(𝕋)函数的所有Lebesgue点处逐点收敛到原函数。
AI 中文摘要
1936年，Zalcwasser证明了三角Fourier部分和平方子序列的算术平均几乎处处收敛，并询问该结果是否可推广到更高次幂及分数阶Cesàro平均。我们以更强的逐点形式对这两个问题给出肯定回答。设0<α≤1，P是次数为正整数且首项系数为正的整数值多项式，我们证明：对所有足够大的k，其第k项等于P(k的任意序列对应的Fourier部分和的(C,α)平均，在所有f∈L¹(𝕋)的每个Lebesgue点x处收敛到f(x)。
英文摘要
In 1936, Zalcwasser proved the almost everywhere convergence of the arithmetic means of the square subsequence of trigonometric Fourier partial sums and asked whether this result extends to higher powers and to Cesàro means of fractional order. We give affirmative answers to both questions in a stronger pointwise form. Let $0<α\leq 1$, and let $P$ be an integer-valued polynomial of degree with positive leading coefficient. We prove that the $(C,α)$ means of the Fourier partial sums along any sequence whose $k$-th term equals $P(k)$ for all sufficiently large $k$ converges to $f(x)$ at every Lebesgue point $x$ of every $f\in L^{1}(\T)$.
