Embedding theorems in constructive approximation

Abstract

In this paper necessary and sufficient conditions for the accuracy of embedding theorems of different function classes are obtained. The main result of the paper is the criterion for embedding between the generalized Weyl-Nikol'skii and Lipschitz classes. To define the Weyl-Nikol'skii classes, we use the concept of a (λ, β)-derivative, which is a generalization of the derivative in the sense of Weyl. As corollaries, we obtain estimates of norms and moduli of smoothness of transformed Fourier series.

Inequalities in Approximation Theory Involving Fractional Smoothness in L p, 0 < p < 1

“…Remark that in the case α ∈ N, Lemma 3.1 was proved in [10]; the case α > 1/p 1 − 1 was considered in [18] and [35].…”

“…Remark that in the case 1 ≤ p < ∞, inequalities of type (2.13) can be found in [45] and [41, p. 154] (see also the general case in [35]).…”

“…where θ = min(2, p) and the constant C is independent of f and δ. Remark that inequality (1.9) can be found, e.g., in [4]; inequality (1.10) is proved in [35] (see also [14] and [12] for α, β ∈ N). It turns out that in the case 0 < p < 1, inequalities (1.9) and (1.10) have been studied only for integer parameters α and β.…”

“…For the first time, the modulus (1.8) appeared in 1970's (see [4, p. 788] and [39]). At present, fractional moduli of smoothness are extensively studied and have several important applications to sharp inequalities of different metrics and embedding theorems (see [18], [32], [33], [35], [36], [43], see also the monograph [27] and the literature therein). It is known that for any f ∈ L p (T), 1 ≤ p < ∞, and α, β > 0, the following two inequalities are fulfilled:…”

Derivatives of a polynomial of best approximation and modulus of smoothness in generalized Lebesgue spaces with variable exponent

“…In addition, relationship between fractional modulus of smoothness of the function and its de la Vallée-Poussin sums is studied. Simillar results in usual Lebesgue spaces have been investigated in [32,35]. Note that, in the proof of the main results we use the method as in the proof of [32,35].…”