Best constants of harmonic approximation on classes associated with the Laplace operator

We compute the best constants of approximation by entire functions of spherical type and by trigonometric polynomials of spherical degree on classes of functions f satisfying the condition Δ^kf_Lp1, where p=1 or 2 and Δ is the Laplace operator.     

Best one-sided approximation of convolution classes by cardinal splines

Let G be a Polya frequency density and its periodization. We get the exact value of upper approximation of S₁ (G), which is a convolution class with kernel G, by cardinal splines corresponding to G. Similar result has been obtained for periodic convolution class with kernel as well. Supported in part by Postdoctral Fellowship Foundation provided by Chinese Academy of Sciences     

On approximation of convolution classes

Asymptotic equalities are found for the least upper bounds of the best approximations of some convolution classes with even kernel in the metric of a spaceL _p.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 26–31, January, 1993.     

Exact estimation of an approximation of some classes of differentiable functions by convolution operators

We obtained explicit formulas for the calculation of a value of the approximation of the classes W ^r,β _p,n by convolution operators of a special form. Here, \( \beta \in \mathbb{Z} \) , r > 0, \( n \in \mathbb{N} \), and p = 1 or p = ∞. As partial cases, we deduced explicit formulas for a value of the approximation of the indicated classes by generalized Abel–Poisson means, biharmonic Poisson operators, and Riesz and Cesáro means. In some cases, we constructed the asymptotic expansions in a parameter for a value of the approximation of the indicated classes. For natural r, some results were obtained by Nikol’skii, Nagy, Timan, Telyakovskii, Baskakov, Falaleev, Kharkevych, and other mathematicians.     

Best one-sided approximation of certain classes of functions

This considers the question of the best one-sided approximation of certain classes of continuous periodic functions by means of trigonometric polynomials of order n-1 in the metric L ₂ ^p (1p<). Precise upper bounds are obtained for the best one-sided approximation of classes of 2/n-periodic functions H_,n [having arbitrary prescribed modulus of continuity (t)] in the metric L ₂ ^p , as well as of classes of 2-periodic functions H [having prescribed modulus of continuity (t) with definite limits] in the metric L ₂ ¹ .Translated from Matematicheskie Zametki, Vol. 10, No. 6, pp. 615–626, December, 1971.The author warmly thanks N. P. Korneichuk for his constant attention to this work.     

Best m-term one-sided trigonometric approximation of some function classes defined by a kind of multipliers

In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm Lp （1 ≤ p ≤ ∞） of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results.     

Two classes of linear equations of discrete convolution type with harmonic singular operators

In this paper, we investigate two classes of linear equations of discrete convolution type with harmonic singular operator. Using the Laurent transform theory, we turn the above linear equations into Riemann boundary value problems. Then, the solutions of the equations are obtained in the class of Hölder continuous functions.     

Best approximation of a differentiation operator in L2-space

This paper contains the magnitude of the best approximation in the L₂-sense of a k-th order differentiation operator of a bounded linear operator A(f) which acts on the class of functions which are differentiable n times.Translated from Matematicheskie Zametki, Vol. 3, No. 2, pp. 157–164, February, 1968.     

On n-widths of some periodic convolution classes in Orlicz spaces

In this paper we extend the some accurate results on n-width of Sobolevclass W_∞~γ in L_p spaces to Orlicz spaces,and establish the correspondingresults for dual cases.     

On some classes of infinitely differentiable operator semigroups

Given a linear relation (a multivalued linear operator), we construct an infinitely differentiable operator semigroup and study its properties.     

Solvability of some classes of nonlinear integro-differential equations with noncompact operator

For a class of nonlinear integrodifferential equations with a noncompact Urysohn-type operator we prove the existence of nonnegative bounded solutions. We study the asymptotic behavior of solutions at infinity. We give some examples that are of practical interest.