Non-symmetric approximations of functional classes by splines on the real line - Parfinovych, N.V. (Carpathian Mathematical Publications, 2021)
Let S_h,m, h>0, mN, be the spaces of polynomial splines of order m of deficiency 1 with nodes at the points kh, kZ. We obtain exact values of the best (α, β)-approximations by spaces S_h,mL_1(R) in the space L_1(R) for the classes W^r_1,1(R), rN, of functions, defined on the whole real line, integrable on R and such that their rth derivatives belong to the unit ball of L_1(R). These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes W^r_1,1(R) by splines from S_h,mL_1(R) (case α=β=1), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes W^r_1(T) of 2π-periodic functions with rth derivative belonging to the unit ball of L_1(T) by periodic polynomial splines of minimal deficiency. As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes W^r_1 by polynomial splines from S_h,m(T). This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes W^r_1 by spaces S_h,m(T).