On the bestL 1-approximations of functional classes by splines under restrictions imposed on their derivatives

We find the exact asymptotics (asn→∞) of the bestL ₁-approximations of classesW ₁ ^r of periodic functions by spliness∈S _{2n, r∼-1} (S _{2n, r∼-1} is a set of 2π-periodic polynomial splines of orderr−1, defect one, and with nodes at the pointskπ/n,k∈ℤ) such that V ₀ ^2π s( ^r-1)≤1+ɛ _n , where {ɛ _n } _n=1 ^∞ is a decreasing sequence of positive numbers such that ɛ _n n ²→∞ and ɛ _n →0 asn→∞. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 435–444, April, 1999.     

On the exact values of the best approximations of classes of differentiable periodic functions by splines

We obtain the exact values of the best L ₁-approximations of classes W ^r F, r ∈ ?, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F, as well as of classes W ^r H ^ω of periodic functions whose rth derivative has a given convex (upward) majorant ω(t) of the modulus of continuity, by subspaces of polynomial splines of order m ≥ r + 1 and of deficiency 1 with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ?, k ∈ ?, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.     

Inequalities of the Kolmogorov type for norms of Riesz derivatives of multivariate functions and some of their applications

We obtained new exact inequalities that estimate the L _?? -norm of the Riesz derivative D ^?? f of a function f defined on $ {\mathbb{R}^m} $ in terms of the uniform norm of the function itself and the L _s -norm of the function acted by the Laplace operator. On a class of functions f such that ||??f||s ?? 1, we solved the problem of approximation of an unbounded operator D ^?? by bounded ones and the problem of optimal recovery of the operator D ^?? on elements of this class given with known error.     

Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities

We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of defect 2.     

Exact values of best approximations for classes of periodic functions by splines of deficiency 2

Mathematical Notes - We obtain exact values of best L 1-approximations for the classes W r F, r ∈ ?, of periodic functions whose rth derivative belongs to a given...     

Kolmogorov type inequalities for norms of Riesz derivatives of multivariate functions and some applications

Let L _∞,s ¹ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that ?f/?x _i ∈ L _s (? ^{m) for each i = 1, ...,m} . New sharp Kolmogorov type inequalities are obtained for the norms of the Riesz derivatives ∥D ^α f∥_∞ of functions f ∈ L _∞,s ¹ (? ^m ). Stechkin’s problem on approximation of unbounded operators D ^α by bounded operators on the class of functions f ∈ L _∞,s ¹ (? ^m ) such that ∥?f∥ _s ≤ 1 and the problem of optimal recovery of the operator D ^α on elements from this class given with error δ are solved.     

Inequalities of the Bernstein type for splines of defect 2

We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order r and defect 2.     

On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

We obtain the exact asymptotics (as n ) of the best L ₁-approximations of classes of periodic functions by splines s S _{2n, r – 1} and s S _{2n, r + k – 1} (S _{2n, r} is the set of 2-periodic polynomial splines of order r and defect 1 with nodes at the points k/n, k Z) under certain restrictions on their derivatives.     

On the order of relative approximation of classes of differentiable periodic functions by splines

In the case where n → ∞, we obtain order equalities for the best L _q -approximations of the classes W _p ^r , 1 ≤ q ≤ p ≤ 2, of differentiable periodical functions by splines from these classes.