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Uniform with respect to the parameter 푎 0 1 estimates of the functions subscript 푓 푎 푥 superscript subscript 푘 1 superscript 푘 푎 푘 푥 and subscript 푔 푎 푥 superscript subscript 푘 1 superscript 푘 푎 푘 푥 by the first terms of their asymptotic expansions subscript 퐹 푎 푥 휋 푎 2 Γ 1 푎 superscript 푥 푎 1 and subscript 퐺 푎 푥 휋 푎 2 Γ 1 푎 superscript 푥 푎 1 are obtained. Namely, it is proved that the inequalities subscript 퐺 푎 푥 푥 2 subscript 푔 푎 푥 subscript 퐺 푎 푥 푥 12 subscript 퐹 푎 푥 휁 푎 휁 3 4 superscript 휋 3 superscript 푥 2 휋 푎 2 subscript 푓 푎 푥 subscript 퐹 푎 푥 휁 푎 1 18 superscript 푥 2 휋 푎 2 are valid for all 푎 0 1 and 푥 0 휋 . It is shown that the estimates are unimprovable in the following sense. In the lower estimate for the sine series, the subtrahend 푥 2 cannot be replaced by 푘 푥 with any 푘 1 2 : the estimate ceases to be fulfilled for sufficiently small 푥 and the values of 푎 close to 1 . In the upper estimate, the subtrahend 푥 12 cannot be replaced by 푘 푥 with any 푘 1 12 : the estimate ceases to be fulfilled for the values of 푎 and 푥 close to 0 . In the lower estimate for the cosine series, the multiplier 휁 3 4 superscript 휋 3 of superscript 푥 2 휋 푎 2 cannot be replaced by any larger number: the estimate ceases to be fulfilled for 푥 and 푎 close to 0 . In the upper estimate for the cosine series, the multiplier 1 18 of superscript 푥 2 휋 푎 2 can probably be replaced by a smaller number but not by 1 24 : for every 푎 0.98 1 , such an estimate would not hold at the point 푥 휋 as well as on a certain closed interval subscript 푥 0 푎 푥 휋 , where → subscript 푥 0 푎 0 as → 푎 limit-from 1 . The obtained results allow us to refine the estimates for the functions subscript 푓 푎 and subscript 푔 푎 established recently by other authors. [ABSTRACT FROM AUTHOR]

For Borel functions on a perfect normal space and, respectively, on a perfect topological space there are two famous convergence Baire classifications: the first one is due to H. Lebesgue and F. Hausdorff and the second one is due to S. Banach. However, for an arbitrary topological space the both classifications are not valid. In this paper the convergence Baire classification of Borel functions on an arbitrary space is given. This classification of Borel functions use new classification of Borel sets. [ABSTRACT FROM AUTHOR]