We introduce the higher order Lipschitz classes Λr(α) and λr(α) of periodic functions by means of the rth order difference operator, where r = 1, 2, ..., and 0 < α ≦ r. We study the smoothness property of a function f with absolutely convergent Fourier series and give best possible sufficient conditions in terms of its Fourier coefficients
in order that f belongs to one of the above classes.
:= [−π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α1,..., αN) and lip (α1,..., αN) for some α1,..., αN > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first
order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients.
Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular
remaining sums of related N-multiple numerical series may be useful in other investigations, too.
We study when sums of trigonometric series belong to given function classes. For this purpose we describe the Nikol’skii class
of functions and, in particular, the generalized Lipschitz class. Results for series with positive and general monotone coefficients
In this paper we introduced the general sequence of linear positive operators via generating functions. Approximation properties of these operators are obtained with the help of the Korovkin Theorem. The order of convergence of these operators computed by means of modulus of continuity Peetre’s K-furictiorial and the elements of the usual Lipschitz class. Also we introduce the r-th order generalization of these operators and we evaluate this generalization by the operators defined in this paper. Finally we give some applications to differential equations.
The main object of this paper is to define the
-Laguerre type positive linear operators and investigate the approximation properties of these operators. The rate of convegence of these operators are studied by using the modulus of continuity, Peetre’s
-functional and Lipschitz class functional. The estimation to the difference |
)| is also obtained for the Meyer-König and Zeller operators based on the
-integers . Finally, the
-th order generalization of the
-Laguerre type operators are defined and their approximation properties and the rate of convergence of this
-th order generalization are also examined.
We describe the functions from Nikol’skii class in terms of behavior of their Fourier coefficients. Results for series with
general monotone coefficients are presented. The problem of strong approximation of Fourier series is also studied.
Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and , 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.