Uniform with Respect to the Parameter a∈(0,1) Two-Sided Estimates of the Sums of Sine and Cosine Series with Coefficients 1/k^a by the First Terms of Their Asymptotics

Uniform with respect to the parameter a∈(0,1) estimates of the functions f_a(x)=∑_k=1^∞k^-acos kx and g_a(x)=∑_k=1^∞k^-asin kx by the first terms of their asymptotic expansions F_a(x)=sin(π a/2)Γ(1-a)x^a-1 and G_a(x)=cos(π a/2)Γ(1-a)x^a-1 are obtained. Namely, it is proved that the inequalities G_a(x)-x21/12 : the estimate ceases to be fulfilled for the values of a and x close to 0 . In the lower estimate for the cosine series, the multiplier ζ(3)/(4π^3) of x^2sin(π a/2) cannot be replaced by any larger number: the estimate ceases to be fulfilled for x and a close to 0 . In the upper estimate for the cosine series, the multiplier 1/18 of x^2sin(π a/2) can probably be replaced by a smaller number but not by 1/24 : for every a∈[0.98,1) , such an estimate would not hold at the point x=π as well as on a certain closed interval x_0(a)≤ x≤π , where x_0(a)→ 0 as a→ 1- . The obtained results allow us to refine the estimates for the functions f_a and g_a established recently by other authors.