54 research outputs found
On approximation methods generated by generalized Bochner-Riesz kernels
Some sharp results related to the convergence of means and families of
operators generated by the generalized Bochner-Riesz kernels are obtained. The
exact order of approximation of functions by these methods via -functional
(or its realization in the case of the space , ) is derived
On moduli of smoothness and averaged differences of fractional order
We consider two types of fractional integral moduli of smoothness, which are
widely used in theory of functions and approximation theory. In particular, we
obtain new equivalences between these moduli of smoothness and the classical
moduli of smoothness. It turns out that for fractional integral moduli of
smoothness some pathological effects arise
Smoothness of functions vs. smoothness of approximation processes
We provide a comprehensive study of interrelations between different measures
of smoothness of functions on various domains and smoothness properties of
approximation processes. Two general approaches to this problem have been
developed: the first based on geometric properties of Banach spaces and the
second on Littlewood-Paley and H\"{o}rmander type multiplier theorems. In
particular, we obtain new sharp inequalities for measures of smoothness given
by the -functionals or moduli of smoothness. As examples of approximation
processes we consider best polynomial and spline approximations, Fourier
multiplier operators on , , , nonlinear
wavelet approximation, etc
Approximation by multivariate Kantorovich-Kotelnikov operators
Approximation properties of multivariate Kantorovich-Kotelnikov type
operators generated by different band-limited functions are studied. In
particular, a wide class of functions with discontinuous Fourier transform is
considered. The -rate of convergence for these operators is given in terms
of the classical moduli of smoothness. Several examples of the
Kantorovich-Kotelnikov operators generated by the -function and its
linear combinations are provided
On weighted conditions for the absolute convergence of Fourier integrals
In this paper we obtain new sufficient conditions for representation of a
function as an absolutely convergent Fourier integral. Unlike those known
earlier, these conditions are given in terms of belonging to weighted spaces.
Adding weights allows one to extend the range of application of such results to
Fourier multipliers with unbounded derivatives
On -error of bivariate polynomial interpolation on the square
We obtain estimates of the -error of the bivariate polynomial
interpolation on the Lissajous-Chebyshev node points for wide classes of
functions including non-smooth functions of bounded variation in the sense of
Hardy-Krause. The results show that -errors of polynomial interpolation on
the Lissajous-Chebyshev nodes have almost the same behavior as the polynomial
interpolation in the case of the tensor product Chebyshev grid
On the growth of Lebesgue constants for convex polyhedra
In the paper, new estimates of the Lebesgue constant for convex polyhedra
are obtained. The main result states that if is a
convex polyhedron such that , then where is a size of the
triangulation of .Comment: accepted in Trans. Amer. Math. So
Sharp estimates of approximation of periodic functions in H\"older spaces
The main purpose of the paper is to study sharp estimates of approximation of
periodic functions in the H\"older spaces for all
and . By using modifications of the classical
moduli of smoothness, we give improvements of the direct and inverse theorems
of approximation and prove the criteria for the precise order of decrease of
the best approximation in these spaces. Moreover, we obtained strong converse
inequalities for general methods of approximation of periodic functions in
Hardy-Littlewood and Ulyanov inequalities
We give the full solution of the following problem: obtain sharp inequalities
between the moduli of smoothness and
for . A similar problem for the
generalized -functionals and their realizations between the couples and is also solved.
The main tool is the new Hardy-Littlewood-Nikol'skii inequalities. More
precisely, we obtained the asymptotic behavior of the quantity where the supremum
is taken over all nontrivial trigonometric polynomials of degree at most
and are the Weyl-type
differentiation operators.
We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces.
Finally, we apply the obtained estimates to derive new embedding theorems for
the Lipschitz and Besov spaces
Inequalities in approximation theory involving fractional smoothness in ,
In the paper, we study inequalities for the best trigonometric approximations
and fractional moduli of smoothness involving the Weyl and Liouville-Gr\"unwald
derivatives in , . We extend known inequalities to the whole range
of parameters of smoothness as well as obtain several new inequalities. As an
application, the direct and inverse theorems of approximation theory involving
the modulus of smoothness , where
is a fractional derivative of the function , are derived. A
description of the class of functions with the optimal rate of decrease of a
fractional modulus of smoothness is given
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