Properties of moduli of smoothness in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e25" altimg="si7.svg"><mml:mrow><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>

2020

Bull. Math. Sci.

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 Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.

“…We will use the following Jackson and Nikolskii-Stechkin inequalities, see, e.g. [62,5.3.2;67,Theorem 3] for the case 1 ≤ p ≤ ∞ and [37] for the case 0 < p < 1:…”

Section: Smoothness Of Best Approximantsmentioning

confidence: 99%

Smoothness of functions versus smoothness of approximation processes

Kolomoitsev

2020

Bull. Math. Sci.

“…On the other hand, to estimate I, we use monotonicity properties (noting that ω α (f, t) p /t α is equivalent to a decreasing function, see, e.g., [44]),…”

Section: Embeddings With Constant Integrabilitymentioning

confidence: 99%

“…Among many other applications of embeddings between Lipschitz spaces we highlight the Ulyanov-type inequalities for moduli of smoothness (see [23]). This topic is a cornerstone in the study of various problems in approximation theory [21,43] and functional analysis [45].…”

Section: Introductionmentioning

confidence: 99%

Embeddings and characterizations of Lipschitz spaces

Domínguez

Haroske

Journal de Mathématiques Pures et Appliquées

2020

Self Cite

In this paper we give a thorough study of Lipschitz spaces. We obtain the following new results: (i) Sharp Jawerth-Franke-type embeddings between the Besov and Lipschitz spaces extending the classical results for Besov and Sobolev spaces. (ii) Sharp embeddings between Lipschitz spaces with different parameters extending the Brézis-Wainger result. (iii) Characterizations for Lipschitz norms via wavelets.

“…is the best approximation of f by trigonometric polynomials of degree at most j, and ω r ( f, δ) L p (T) is the classical r th modulus of smoothness. Sharp Jackson and inverse approximation inequalities were further developed in many papers (see for example [8][9][10]15,17,31,54] and the references therein). Our proof of Theorem 1.1 is based on the corresponding Littlewood-Paley decomposition in the Dunkl setting; cf.…”

Section: Sharp Jackson and Inverse Inequalitiesmentioning

confidence: 99%

“…3.1] (k ≡ 0) and [21] (k(•) ≥ 0). For the case of fractional moduli, see [31,47] (k ≡ 0). The discussion on various ways to define moduli of smoothness can be found in [21,Sec.…”

Section: Proposition 34 ([18]mentioning

confidence: 99%

Sharp approximation theorems and Fourier inequalities in the Dunkl setting

Горбачев

Ivanov

Journal of Approximation Theory

2020

In this paper we study direct and inverse approximation inequalities in L p (R d ), 1 < p < ∞, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish new estimates of the modulus of smoothness of a function f via the fractional powers of the Dunkl Laplacian of approximants of f . Moreover, we obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl transforms. Needed Pitt-type and Kellogg-type Fourier-Dunkl inequalities are derived. c