Best Approximations Of Periodic Functions Research Articles

Best approximation of periodic functions in the Lebesgue spaces

Best approximation of periodic functions in the Lebesgue spaces

Generalized Smoothness and Approximation of Periodic Functions in the Spaces Lp, 1 < p < +∞

Norms of images of operators of multiplier type with an arbitrary generator are estimated by using best approximations of periodic functions of one variable by trigonometric polynomials in the scale of the spaces Lp, 1 < p < +∞. A Bernstein-type inequality for the generalized derivative of the trigonometric polynomial generated by an arbitrary generator ψ, sufficient constructive ψ-smoothness conditions, estimates of best approximations of ψ-derivatives, estimates of best approximations of ψ-smooth functions, and an inverse theorem of approximation theory for the generalized modulus of smoothness generated by an arbitrary periodic generator are obtained as corollaries.

Multiple Modules of Continuity and the Best Approximations of Periodic Functions in Metric Spaces

It is proved that, under the condition $$ {M}_{\varPsi}\left(\frac{1}{2}\right)<1 $$ , where MΨ is a stretching function Ψ in the space LΨ, the Jackson inequalities $$ \underset{n}{\sup}\underset{f\in {L}_{\varPsi },f\ne \mathrm{const}}{\sup}\frac{E_{n-1}{(f)}_{\varPsi }}{\omega_k{\left(f,\frac{\pi }{n}\right)}_{\varPsi }}<\infty $$ are true; here, En−1(f)Ψ is the best approximation of f by trigonometric polynomials of degree at most n − 1 and ωk(f, h)Ψ is the module of continuity of f of order k, k ∈ N. We study necessary and sufficient conditions for the function f under which the following relation is true: En − 1(f)Ψ $$ {\omega}_k{\left(f,\frac{\pi }{n}\right)}_{\varPsi } $$ .

On the constant and step in Jackson’s inequality for best approximations by trigonometric polynomials and by Haar polynomials

Two sharp results for best approximations of periodic functions are established in this paper. We prove the sharpness of the step of the modulus of continuity in Jackson’s inequality with least possible constant for approximations by trigonometric polynomials. We also prove the sharpness of the constants in a Jackson-type inequality for approximations by Haar polynomials in several variables.

On Jackson Type Inequalities for the Best Approximations of Periodic Functions

UDC 517.5 We develop a new approach to the proof of 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. The proposed approach is based on approximation methods constructed by using Fejer and Jackson–Vallee - Poussin classical kernels. Bibliography :1 4titles.

On the Constants in Generalized Jackson-Type Inequalities

Some modifications of Jackson-type inequalities are established, including estimates on the best approximations of periodic functions by trigonometric polynomials in terms of the moduli of continuity of higher orders. Special attention is paid to bounding the constants occurring in the inequalities obtained.

Estimates for the Best Approximations of Periodic Functions in Terms of Deviations

We estimate the best approximation of a function by trigonometric polynomials in terms of deviations. To illustrate general results, we consider estimates connected with the Vallee–Poussin integral and the generalized Weierstrass integral.

Inequalities of the type of the generalized Jackson theorem for the best approximations

Very simple methods for deriving upper estimates of the type of the generalized Jackson theorem for the best approximations of periodic functions are proposed, The results obtained are similar to the generalized Jackson theorem, providing estimates in terms of the moduli of continuity. Bibliography: 12 titles.

Best approximations of periodic functions in generalized lebesgue spaces

In generalized Lebesgue spaces with variable exponent, we determine the orders of the best approximations in the classes of (ψ; β)-differentiable 2π-periodic functions, deduce an analog of the well-known Bernstein inequality for the (ψ; β)-derivative, and apply this inequality to prove the inverse theorems of approximation theory in these classes.

Best approximation of periodic functions of several variables from the classes $ MB_{{p,\theta }}^{\omega } $

We obtain exact-order estimates for the best approximation of periodic functions of several variables from the classes $ MB_{{p,\theta }}^{\omega } $ by trigonometric polynomials with the “numbers” of harmonics from graded hyperbolic crosses in the metric of the space L q :

Approximation of functions on the Sobolev space with a Gaussian measure

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Vallee-Poussin operators, Ces`aro operators, Abel operators, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the Lq space for 1 ⩽ q < ∞, and the Fourier partial summation operators and the Vallee-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the Lq space for 1 ⩽ q < ∞.

Best approximations of periodic functions of several variables from the classes B p, θ Ω

We obtain order-sharp estimates of best approximations for the classes Bp, θΩ of periodic functions of several variables by trigonometric polynomials whose spectra are generated by the level surfaces of the function Ω(t).

Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of L p

We obtain upper bounds in terms of Fourier coefficients for the best approximation by an “angle” and for norms in the metric of L p for functions of two variables defined by trigonometric series with coefficients such that $$a_{l_1 l_2 } \to 0$$ as l 1 + l 2 → ∞ and $$\mathop \sum \limits_{k_1 = 0}^\infty \mathop \sum \limits_{k_2 = 0}^\infty \left( {\mathop \sum \limits_{l_1 = k_1 }^\infty \mathop \sum \limits_{l_2 = k_2 }^\infty \left| {\Delta ^{12} a_{l_1 \;l_2 } } \right|} \right)^p (k_1 + 1)^{p - 2} \;(k_2 + 1)^{p - 2} < \infty$$ for a certain p, 1 < p < ∞.

On the best approximation of periodic functions of two variables by polynomial splines

We consider the problem of the best approximation of periodic functions of two variables by a subspace of splines of minimal defect with respect to a uniform partition.

Estimates of the best approximations of periodic functions by trigonometric polynomials in terms of averaged differences and the multidimensional Jackson's theorem

In the first section the best approximations of periodic functions of one real variable by trigonometric polynomials are studied. Estimates of these approximations in terms of averaged differences are obtained. A multidimensional generalization of these estimates is presented in the second section. As a consequence. The multidimensional Jackson's theorem is proved.

Best approximations of differentiable functions in the metric of the space L2

The relation between the best approximations of periodic functions by trigonometric polynomials and the moduli of continuity of their arbitrary derivatives in the metric of the space L2 is studied.

Best approximation of periodic functions by trigonometric polynomials in L2

Estimates are gotten for the best approximations in L2 (0, 2π) of a periodic function by trigonometric polynomials in terms of its m-th continuity modulus or in terms of the continuity modulus of its r-th derivative. The inequality En−1(f)L2<(C 2m m ωm(2π/n;f)L2(1≠ const) is proved, where the constant (C 2m m )−1/2 is unimprovable for the whole space L2 (0, 2π). Two titles are cited in the bibliography.