Priblizhenie ψ-differentsiruemykh funktsii kratnymi summami Fur'e v integral'noi metrike

Izuchayutsya ukloneniya pryamougol'nykh chastnykh summ Fur'e na mnozhestvakh -differentsiruemykh funktsii mnogikh peremennykh v integral'noi metrike.     

Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary

We establish estimates for groups of deviations of Faber series in closed domains with piecewise-smooth boundary.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 187–197, February, 2005.     

Direct and inverse theorems on approximation of functions defined on a sphere in the space <Emphasis Type="Italic">S</Emphasis><Superscript>(<Emphasis Type="Italic">p,q</Emphasis>)</Superscript>(σ<Superscript>m</Superscript>)

We prove direct and inverse theorems on the approximation of functions defined on a sphere in the space S ^(p,q)(σ^m), m ≥ 3, in terms of the best approximations and moduli of continuity. We consider constructive characteristics of functional classes defined by majorants of the moduli of continuity of their elements. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 901–911, July, 2007.     

Summation of Fourier-Laplace Series in the Space <Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>)

We establish estimates for the rate of convergence of a group of deviations on a sphere in the space L(S ^m ), m ≥ 3. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 496–504, April, 2005.     

Strong summability of orthogonal expansions of summable functions. II

We study the problem of strong summability of Fourier series in orthonormal systems of polynomialtype functions and establish local characteristics of the points of strong summability of series of this sort for summable functions. It is shown that the set of these points is a set of full measure in the region of uniform boundedness of the systems under consideration.     

Estimates for a Group of Deviations in Generalized Hölder Metric

We present order relations for a group of deviations of a function f(·) H in terms of partial Fourier sums of this function in a generalized Hölder metric defined in a generalized Hölder space H _* H .     

Marcinkiewicz-type strong means of Fourier—Laplace series

We obtain estimates for Marcinkiewicz-type strong means of the Fourier—Laplace series of continuous functions in terms of the best approximations.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 763–773, June, 2004.     

Strong summability of orthogonal expansions of summable functions. I

We study the problem of strong summability of Fourier series in orthonormal systems of polynomial-type functions and establish local characteristics of the points of strong summability of series of this sort for summable functions. It is shown that the set of these points is a set of full measure in the region of uniform boundedness of systems under consideration.     

(ϕ, α)-Strong Summability of Fourier–Laplace Series for Functions Continuous on a Sphere

We establish upper bounds for approximations by generalized Totik strong means applied to deviations of Cezàro means of critical order for Fourier–Laplace series of continuous functions. The estimates obtained are represented in terms of uniform best approximations of continuous functions on a unit sphere.     

Multiple Fourier sums and ϕ-strong means of their deviations on the classes of $$\bar \psi $$ -differentiable functions of many variables

We present the results concerning the approximation of -differentiable functions of many variables by rectangular Fourier sums in uniform and integral metrics and establish estimates for the ϕ-strong means of their deviations in terms of the best approximations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1075–1093, August, 2007.