Cesaro Means along Polynomial Subsequences of Fourier Partial Sums at Lebesgue Points

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 $0<\alpha \le 1$, and let $P$ be an integer-valued polynomial of degree with positive leading coefficient. We prove that the $(C,\alpha )$ 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 $f\in L^1(\text{T})$.