AMS Subject Classification 2010: 41A25, 41A35, 41A40, 41A63, 41A65, 42A38, 42A85, 42B10, 42B20
The paper presents a description of the optimal rate of approximation as well as of a broad class of functions that possess it for convolution operators acting in the so-called homogeneous Banach spaces of functions on Rd. The description is the same in any such space and uses the Fourier transform. Simple criteria for establishing upper estimates of the approximation error via a K-functional...

AMS Subject Classification 2010: 41A25, 41A27, 41A35, 41A36, 41A40, 42Al6, 42A85.
The paper is concerned with establishing direct estimates for convolution operators on homogeneous Banach spaces of periodic functions by means of appropriately defined Kfunctional. The differential operator in the K-functional is defined by means of strong limit and described explicitly in terms of its Fourier coefficients. The description is simple and independent of the homogeneous Banach space. Saturation...

2000 Mathematics Subject Classification: 46B70, 41A25, 41A17, 26D10.
∗Part of the results were reported at the Conference “Pioneers of Bulgarian Mathematics”,
Sofia, 2006.
Certain types of weighted Peetre K-functionals are characterized by means
of the classical moduli of smoothness taken on a proper linear
transforms of the function. The weights with power-type asymptotic at the
ends of the interval with arbitrary real exponents are considered. This paper
extends the method and results...

2000 Mathematics Subject Classification: 46B70, 41A10, 41A25, 41A27, 41A35, 41A36,
42A10.
The paper presents a method of relating two K-functionals by
means of a continuous linear transform of the function. In particular, a
characterization of various weighted K-functionals by unweighted fixed-step
moduli of smoothness is derived. This is applied in estimating the rate of
convergence of several approximation processes.
Partially supported by grant No. 103/2007 of the National...

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