We modify certain results obtained by I. A. Ibragimov concerning the remainder term in the Central Limit Theorem and the remainder term in the approximation of the distribution function for normed sums of i.i.d. rv's by a portion of the Chebyshev series and we then use these to complete some results appearing in a joint article by C. C. Heyde and the author (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 21 255-268) (as well as in a similar article by F. N. Galstian (Theor. Probability Appl. 16 3, 528-533)).
"Generalisation and Application of Some Results of Ibragimov on Convergence to Normality." Ann. Probab. 3 (5) 897 - 902, October, 1975. https://doi.org/10.1214/aop/1176996279