Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials

We establish lower and upper bounds for the quantity

Shape-Preserving Smoothing of 3-Convex Splines of Degree 4

For every 3-convex piecewise-polynomial function s of degree ≤4 with n equidistant knots on [0, 1] we construct a 3-convex spline s ₁ (s ₁ ∈ C ⁽³⁾) of degree ≤4 with the same knots that satisfies the inequality

Cayley transform of the generator of a uniformly bounded <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup of operators

We consider the problem of estimates for the powers of the Cayley transform V = (A + I)(A - I)^–1 of the generator of a uniformly bounded C ₀-semigroup of operators e ^tA , t 0, that acts in a Hilbert space H. In particular, we establish the estimate . We show that the estimate is true in the following cases: (a) the semigroups e ^tA and are uniformly bounded; (b) the semigroup e ^tA uniformly bounded for t is analytic (in particular, if the generator of the semigroup is a bounded operator).Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1018–1029, August, 2004.     

Best Polynomial Approximations in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> and Widths of Some Classes of Functions

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions f(x) ∈ L ₂ ^r (r ∈ ℤ₊) and expressions containing moduli of continuity of the kth order ω_k(f(^r), t). Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from L ₂. For the classes (k, r, Ψ_*) defined by ω _k(f ^(r), t) and the majorant , we determine the exact values of different widths in the space L₂.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1458–1466, November, 2004.     

A remark concerning the modulus of smoothness introduced by Ditzian and Totik

For each functionf(x) continuous on the segment [–1, 1], we set . We study the relationship between the ordinarykth modulus of continuity of therth derivative of the function and thekth modulus of continuity with weight _r of the rth derivativef ^(r) of the functionf introduced by Ditzian and Totik. Thus, ifr is odd andk is even, we prove that these moduli are equivalent ast0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1627–1638, December, 1995.     

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.     

Complete Asymptotics of the Deviation of a Class of Differentiable Functions from the Set of Their Harmonic Poisson Integrals

On a class of differentiable functions W ^r and the class of functions conjugate to them, we obtain a complete asymptotic expansion of the upper bounds of deviations of the harmonic Poisson integrals of the functions considered.     

Multidimensional Lagrange–Yen-Type Interpolation Via Kotel'nikov–Shannon Sampling Formulas

Direct finite interpolation formulas are developed for the Paley–Wiener function spaces and , where contains all bivariate entire functions whose Fourier spectrum is supported by the set = Cl{(u, v) |u| + |v| < ], while in the Fourier spectrum support set of its d-variate entire elements is [–, ] ^d . The multidimensional Kotel'nikov–Shannon sampling formula remains valid when only finitely many sampling knots are deviated from the uniform spacing. By using this interpolation procedure, we truncate a sampling sum to its irregularly sampled part. Upper bounds of the truncation error are obtained in both cases.According to the Sun–Zhou extension of the Kadets -theorem, the magnitudes of deviations are limited coordinatewise to . To avoid this inconvenience, we introduce weighted Kotel'nikov–Shannon sampling sums. For , Lagrange-type direct finite interpolation formulas are given. Finally, convergence-rate questions are discussed.     

Smooth Solution of the Dirichlet Problem for a Quasilinear Hyperbolic Equation of the Second Order

On the basis of the exact solution of the linear Dirichlet problem , we obtain conditions for the solvability of the corresponding Dirichlet problem for the quasilinear equation u _tt – u _xx = f(x, t, u, u _t).     

Criteria for the Well-Posedness of the Cauchy Problem for Differential Operator Equations of Arbitrary Order

In Banach spaces, we investigate the differential equation with closed linear operators A _j (generally speaking, the operator coefficient A _n of the higher derivative is degenerate). We obtain well-posedness conditions that characterize the continuous dependence of solutions and their derivatives on initial data. Abstract results are applied to partial differential equations.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1484–1500, November, 2004.     

Best n-Term Approximations in Spaces with Nonsymmetric Metric

We determine exact values of n-term approximations of q-ellipsoids in the spaces .     

On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space . The perturbed operators are defined by the Krein resolvent formula , Im z 0, where B _z are finite-rank operators such that dom B _z dom A = |0}. For an arbitrary system of orthonormal vectors satisfying the condition span | _i } dom A = |0} and an arbitrary collection of real numbers , we construct an operator that solves the eigenvalue problem . We prove the uniqueness of under the condition that rank B _z = n.     

Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p [1, ], the following unimprovable inequality holds for functions :

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Let = (₁,...,_d) be a vector with positive components and let D be the corresponding mixed derivative (of order _j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that

Rate of Convergence of Positive Series

We investigate the rate of convergence of series of the form

On Saturation of Linear Summation Methods for Fourier Series in the Spaces $$S_\varphi ^p $$

We consider the problem of the saturation of linear summation methods for Fourier series in the spaces , p > 0. We show that the saturation of a linear method and the saturation order are independent of the parameters X, , and p that define the space (X).     

Approximation of $$\bar \omega $$ -integrals of continuous functions defined on the real axis by Fourier operators

We obtain asymptotic formulas for the deviations of Fourier operators on the classes of continuous functions and in the uniform metric. We also establish asymptotic laws of decrease of functionals characterizing the problem of the simultaneous approximation of -integrals of continuous functions by Fourier operators in the uniform metric.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 663–676, May, 2004.     

Criterion of Polynomial Denseness and General Form of a Linear Continuous Functional on the Space C w 0

For an arbitrary function , we determine the general form of a linear continuous functional on the space C _w ⁰ . The criterion for denseness of polynomials in the space established by Hamburger in 1921 is extended to the spaces C _w ⁰ .     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

Order comparison of norms of polynomials in regions of the complex plane

For an arbitrary Jordan region G, inequalities are obtained which relate the norm with the norms and of a polynomial P_n(z) of degree n.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1277–1280, September, 1991.