On the moduli of continuity and fractional-order derivatives in the problems of best mean-square approximations by entire functions of the exponential type on the entire real axis - Vakarchuk, S. B. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
The exact Jackson-type inequalities with modules of continuity of a fractional order α (0,∞ ) are obtained on the classes of functions defined via the derivatives of a fractional order α (0,∞ ) for the best approximation by entire functions of the exponential type in the space L_2(R). In particular, we prove the inequality 2^- β /2σ^- α (1 - t)^- β /2 ≤ { A_σ (f) / ω_β (D^α f, t/σ ) : f L^α_2 (R)} ≤ σ^-α (1/t^2 + 1/2)^β /2, where β [1,∞ ), t (0, π ], σ (0,∞ ). The exact values of various mean u -widths of the classes of functions determined via the fractional modules of continuity and majorant satisfying certain conditions are also determined.