Metric Of L1 Research Articles

Linear approximation methods and the best approximations of the Poisson integrals of functions from the classes $ {H_{{\omega_p}}} $ in the metrics of the spaces L p

We obtain upper estimates for the least upper bounds of approximations of the classes of Poisson integrals of functions from $ {H_{{\omega_p}}} $ for 1 ≤ p < ∞ by a certain linear method U n * in the metric of the space L p . It is shown that the obtained estimates are asymptotically exact for p = 1: In addition, we determine the asymptotic equalities for the best approximations of the classes of Poisson integrals of functions from $ {H_{{\omega_1}}} $ in the metric of the space L 1 and show that, for these classes, the method U n * is the best polynomial approximation method in a sense of strong asymptotic behavior.

On the guaranteed accuracy of a dynamical recovery procedure for controls with bounded variation in systems depending linearly on the control

We obtain an upper bound for the accuracy of a modification of the finite-step dynamical reconstruction algorithm proposed by Yu. S. Osipov and A. V. Kryazhimskii for the unknown control in the dynamical system from inaccurate data on its trajectory We establish that, for the case in which the control is a function of bounded variation and the subspace of the image of the matrix of its coefficients has constant dimension, the asymptotic order in the metric of the space L1[a, b] of its accuracy with respect to the input error is 1/2.

The strong L 1-greedy property of the Walsh system

For any 0 1 − ɛ such that for each function f(x) e L1 (0, 1) a function g(x) e L1 (0, 1) exists such that it coincides with f (x) on E, its Fourier—Walsh series converges to it in the metric of L1 (0, 1), and all nonzero terms of the sequence of Fourier coefficients of the new function obtained by the Walsh system have the modulo decreasing order; consequently, the greedy algorithm for this function converges to it in the L1 (0, 1)-norm.

The Singularity Property of Banach Function Spaces and Unconditional Convergence in L1[0, 1

Let X be a Banach function space, L∞ [0, 1] ⊂ X ⊂ L1[0, 1]. It is proved that if dual space of X has singularity property in closed set E ⊂ [0, 1] then: 1) there exists no orthonormal basis in C[0, 1], which forms an unconditional basis in X in metric of L1[0, 1] space, 2) for the Hardy-Littlewood maximal operator M we have Open image in new window

Entropy Solutions to a Second Order Forward-Backward Parabolic Differential Equation

We prove that the first boundary value problem for a second order forward-backward parabolic differential equation in a bounded domain G T ⊂ ℝd+1, where d ≥ 2, has a unique entropy solution in the sense of F. Otto. Under some natural restrictions on the boundary values this solution is constructed as the limit with respect to a small parameter of a sequence of solutions to Dirichlet problems for an elliptic differential equation. We also show that the entropy solution is stable in the metric of L1(G T ) with respect to perturbations of the boundary values in the metric of L1(∂G T ).

Approximation of Infinitely Differentiable Periodic Functions by Interpolation Trigonometric Polynomials in Integral Metric

We obtain asymptotic equalities for the upper bounds of approximations of periodic infinitely differentiable functions by interpolation trigonometric polynomials in the metric of L1 on the classes of convolutions.

Approximation of Periodic Functions of High Smoothness by Interpolation Trigonometric Polynomials in the Metric of L1

We establish an asymptotically exact estimate for the error of approximation of ℝ2-periodic functions of high smoothness by interpolation trigonometric polynomials in the metric of L1.

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB ρ , 0 ≤ ρ < 1, 1≤p ≤ ∞, of 2π-periodic functions of the form f(t)=u(ρ,t), whereu (ρ,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel Kρ(t) of the convolution f= Kρ *g, ∥g∥ρ≤l, with respect to the metric of L1. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.

Nonsymmetric approximations of a class by a class that is given with the help of a linear differential operator

We obtain certain analogs of comparison theorems for σ-rearrangements of functions from nonsymmetric classes of functions, given with the help of a linear-differential operator, and also obtain inequalities for the best (α, β)-approximation of the class Wℬ Hω by another class of functions in the metric of L1.

Local limlt theorems for functionals of random fields

One investigates convergence for functionals of random fields in the metric of L1.

Distance in the metric of L1 of the distribution of a sum of weakly dependent random variables from the normal distribution function

Distance in the metric of L1 of the distribution of a sum of weakly dependent random variables from the normal distribution function

Best-possible one-sided approximations in the classes 688-01688-01688-01by trigonometric polynomials in the metric of L1

The quantities Open image in new window are calculated, where Open image in new window is the best approximation from above of the function f by trigonometric polynomials of order ⩽n—1 in the metric of L1.

Zolotarev problem in the metric of L1([?1, 1

In this paper we solve the problem of the determination of a polynomial of degree n with given two leading coefficients which has the least deviation from zero in the metric of L1 ([−1, 1]). The extremal polynomial is expressed in the form of some linear combination of Chebyshev polynomials of the second kind.

Approximation in the metric of 𝐿¹(𝑋,𝜇)

Approximation in the metric of 𝐿¹(𝑋,𝜇)