Constants In Jackson-type Inequalities Research Articles

Direct and inverse approximation theorems of functions in the Musielak-Orlicz type spaces

In Musilak-Orlicz type spaces ${\mathcal S}_{\bf M}$, direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.

Exact constants in Jackson-type inequalities for the best mean square approximation in L 2(ℝ) and exact values of mean 𝜈-widths of the classes of functions

On the classes of functions $$ {L}_2^r\left(\mathbb{R}\right) $$ , where r ∈ℤ + , for the characteristics of smoothness $$ {\Lambda}_k\left(f,t\right)={\left\{\left(1/t\right){\int}_0^t\left\Vert {\varDelta}_h^k(f)\left\Vert {}^2\right. dh\right.\right\}}^{\kern0em 1/2},t\in \left(0,\infty \right),k\in \mathbb{N} $$ , the exact constants in the Jackson-type inequalities have been obtained in the case of the best mean square approximation by entire functions of the exponential type in the space L 2(ℝ). The exact values of mean 𝜈-widths of the classes of functions defined by Λ k (f) and the majorants Ψ satisfying some conditions are calculated.

Inequalities between best polynomial approximations and some smoothness characteristics in the space L 2 and widths of classes of functions

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λk(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L2. On the basis of this, for differentiable functions in the classes L2r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ωk. For classes of functions defined by their smoothness characteristics Λk(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.

Generalized smoothness characteristics in Jackson-type inequalities and widths of classes of functions in L 2

In the space L2, we study a number of smoothness characteristics of functions; our study is based on the use of the generalized shift operator τh. For the case inwhich τ is the Steklov operator S, we obtain exact constants in Jackson-type inequalities for some classes of 2π-periodic functions. We also calculate the exact values of the n-widths of function classes defined by the smoothness characteristics under consideration.

Constants in Jackson Type Inequalities for the Best Approximation of Periodic Functions Such that Some of Their Fourier Coefficients Vanish

We consider the class of continuous 2π-periodic functions such that some of their Fourier coefficients vanish. For such functions we study constants in a generalized Jackson theorem providing an estimate for the best approximation by trigonometric polynomials with the help of the moduli of continuity of an arbitrary order.

On some extremal problems of approximation theory of functions on the real axis II

The work is devoted to the solution of a number of extremal problems of approximation theory of functions on the real axis \( \mathbb{R} \). In the space L 2(\( \mathbb{R} \)), the exact constants in Jackson-type inequalities are calculated. The exact values of average ν-widths are obtained for the classes of functions from L 2(\( \mathbb{R} \)) that are defined by averaged k-order moduli of continuity and for the classes of functions defined by K-functionals. In the chronological order, the sufficiently complete analysis of the final results related to the solution of extremal problems of approximation theory in the periodic case and on the whole real axis is carried out.

Best mean-square approximation of functions defined on the real axis by entire functions of exponential type

Exact constants in Jackson-type inequalities are calculated in the space L 2.(ℝ) in the case where the quantity of the best approximation A σ (f) is estimated from above by the averaged smoothness characteristic $$ {\varPhi_2}\left( {f,t} \right)=\frac{1}{t}\int\limits_0^t {\left\| {\Delta_h^2(f)} \right\|dh} $$ . We also calculate the exact values of the average v-widths of classes of functions defined by Φ2. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 604–615, May, 2012.

Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in L 2

We find the exact values of the n-widths for the classes of periodic differentiable functions in L2[0, 2π] satisfying the constraint $$\int\limits_0^h {t\tilde \Omega _m^{1/m} (f^{(r)} ;t)dt \leqslant \Phi (h)} ,$$ where h > 0, m ∈ ℕ, r ∈ ℤ+, \(\tilde \Omega _m \)(f(r); t) is the generalized mth order continuity modulus of the derivative f(r) ∈ L2[0, 2π], while Φ(t) is an arbitrary increasing function such that Φ(0) = 0.

The rate of decrease of constants in Jackson type inequalities in dependence of the order of the continuity modulus

Jackson type inequalities for moduli of continuity of arbitrary order are established with the use of linear approximation methods. The constants are smaller than known previously. The results hold in different spaces of periodic and nonperiodic functions. Bibliography: 19 titles.

Exact constants in Jackson-type inequalities for L 2-approximation on an axis

We investigate exact constants in Jackson-type inequalities in the space L 2 for the approximation of functions on an axis by the subspace of entire functions of exponential type.

Exact Constants in Jackson-type Inequalities and Exact Values of Widths

Exact Constants in Jackson-type Inequalities and Exact Values of Widths

Jackson-Type Inequalities in the Space Sp

In the case of approximation of periodic functions in the space Sp, we determine the exact constants in Jackson-type inequalities for the Zygmund, Rogosinski, and de la Vallee Poussin linear summation methods.

On the Best Polynomial Approximations of 2π-Periodic Functions and Exact Values of n-Widths of Functional Classes in the Space L2

To solve extremal problems of approximation theory in the space L2, we use τ-moduli introduced by Ivanov. We determine the exact values of constants in Jackson-type inequalities and the exact values of n-widths of functional classes determined by these moduli.

On Best Polynomial Approximations in L 2

For the τ-moduli of smoothness of mth order, we calculate exact constants in Jackson-type inequalities. We also obtain the exact values of the n-widths of classes of functions whose rth derivatives are characterized by τ-moduli of smoothness majorized by functions satisfying certain constraints. We present an example of the majorant for which all the stated requirements are satisfied.

Widths of classes fromL 2[0, 2π] and minimization of exact constants in Jackson-type inequalities

The problem of minimizing the constant in Jackson-type inequalities with respect to all subspaces of dimensionN, i.e., with respect to the entire set of approximating subspaces of dimensionN, is solved. It is shown that the constant in question is equal to the value of different widths of classL 2 (δ, τ, β, γ, θ).

On exact constants in Jackson-type inequalities in the space L2

The exact dependence of constants in Jackson-type inequalities on the rate of convergence of the Diophantine approximations of certain numbers is obtained.

Exact constant in the Jackson inequality in the space L2

We investigate the lowest (exact) constants in Jackson-type inequalities with the second-order modulus of continuity for the case of approximation of differentiable periodic functions by trigonometric polynomials in the space L2.