Remarks on Wiener's and Blackwell's Theorems

The paper consists of two parts. In the first part conditions are discussed, for which an analytical transformation of a function, presented as an absolutely convergent Fourier series, retains asymptotic behavior of its coefficients. In the second part these conditions are used for seeking the asymptotic behavior of the remainder term in Blackwell's limit theorem. Some statements are also made therein describing the behavior of conditional mean of the first passage time in the random walk with jumps which on the average are negative.