Sergey B. Vakarchuk, Meansquare Approximation of Function Classes, Given on the all Real Axis R by the Entire Functions of Exponential Type, Volume 6, International Journal of Advanced Research in Mathematics (Volume 6)
https://www.scipress.com/IJARM.6.1
Abstract:
    <i>K</i>-functionals <i>K</i> (<i>f</i>,<i> t</i>, <i>L</i><sub>2</sub>(R), <i>L</i><sub>2</sub><i><sup>β</sup></i>(R), which defined by the fractional derivatives of order <i>β</i>&gt;0, have been considered in the space <i>L</i><sub>2</sub>(R). The relation <i>K</i> (<i>f</i>, <i>t<sup>β</sup></i>, <i>L</i><sub>2</sub>(R), <i>L</i><sub>2</sub><i><sup>β</sup></i>(R) ≈ <i>ω<sub>β </sub></i>(<i>f, t</i>) (<i>t</i>&gt;0) was obtained in the sense of the weak equivalence, where ωω<sub>β </sub>(<i>f, t</i>) is the module of continuity of the fractional order <i>β</i> for a function <i>f </i>є <i>L</i><sub>2</sub>(R). Exact values of the best approximation by entire functions of exponential type <i>v</i>∏, <i>v</i> є (0, ∞) have been computed for the classes of functions, given by the indicated <i>K</i>-functionals and majorants Ψ satisfying specific restriction. Kolmogorov, Bernsteinand linear mean <i>v</i>-widths were obtained for indicated classes of functions.
Keywords:
    Best Approximations, Entire Function of Exponential Type, <i>K</i>-Functional, Majorant, Mean<i> v</i>-Width