Sharp Ul'yanov inequalities for generalized Liouville-Weyl derivatives
type: Item
instance of: scholarly article
title: Sharp Ul'yanov inequalities for generalized Liouville-Weyl derivatives
zbMATH Open document ID: 1399.41018
author: Juan-Miguel Gracia
published in: Analysis Mathematica
publication date: 2018-02-13T00:00:00Z
review text: The main focus of the article is sharp \(\left(p,q\right)\)-inequalities between fractional moduli of smoothness. Inequalities of this type are called Ul'yanov-type inequalities. The author considers these inequalities for the generalized Liouville-Weyl derivatives defined in the following way. Let the Fourier series of \(f\in L_1\left({\mathbb T}\right)\) be given by \[ f\left(x\right)\sim\sum_{\nu\in{\mathbb Z}}\hat f_\nu e^{i\nu x}, \] where \(\hat f_\nu\) are the Fourier coefficients of \(f\). The generalized Liouville-Weyl derivative is defined by (provided the right-hand side is a Fourier series of an integrable function) \[ D^\lambda f\left(x\right)\sim\sum_{\nu\in{\mathbb Z}}\lambda_\nu\hat f_\nu e^{i\nu x}, \] where \(\lambda_\nu=\lambda\left(\left| \nu\right| \right)\), for some non-decreasing function \(\lambda:\,\left[0,\infty\right)\to\left[0,\infty\right)\). The author cites several Ul'yanov-type results for these derivatives valid when \(\lambda\) satisfies certain properties and notices at the same time that those results do not cover limiting cases (when \(p=1\), or \(q=\infty\)). The present article proves more general analogues of the aforementioned results, in particular, considering monotonicity requirement replaced by the general monotonicity. Here, a sequence \(\left\{\lambda_n\right\}_{n=1}^\infty\) is said to be general monotone if the relation \[ \sum_{k=n}^{2n}\left| \lambda_k-\lambda_{k+1}\right| \leq C\left| \lambda_n\right| \] holds for all \(n\in{\mathbb N}\) with the constant \(C\) independent of \(n\). Furthermore, the limiting cases are also covered. Let us cite one of the results obtained. Theorem. Let \(f\in L_p\), \(1=p<q<\infty\), \(\theta=1-1/q\). Let the sequence \(\left\{\lambda_n\right\}_{n=1}^\infty\) be general monotone. Then, for any \(\alpha>0\) and \(0<\varepsilon\leq\min\left(\rho,\theta\right)\), \[ \omega_\alpha\left(D^\lambda f,\frac{1}{2^n}\right)_q\lesssim \left(\sum_{m=n}^\infty\left(2^{m\left(\theta+\rho\right)}…
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 42A10
zbMATH DE Number: 6837907
zbMATH Keywords: Ul'yanov-type inequality
zbMATH Keywords: modulus of smoothness
zbMATH Keywords: general monotone sequence
zbMATH Keywords: Liouville-Weyl derivative
zbMATH Keywords: fractional derivative
reviewed by: Alexander Tovstolis
MaRDI profile type: Publication
full work available at URL: s10476-017-0308-0
OpenAlex ID: W2744367423
cites work: Best Trigonometric Approximation, Fractional Order Derivatives and Lipschitz Classes
cites work: Q4273944
cites work: Ul'yanov and Nikol'skii-type inequalities
cites work: Moduli of smoothness of functions and their derivatives
cites work: Liouville-weyl derivatives, best approximations, and moduli of smoothness
cites work: A concept of general monotonicity and applications
cites work: Sharp Ul'yanov-type inequalities using fractional smoothness
cites work: Embedding theorems in constructive approximation
cites work: Q3414500
cites work: Q4072268
cites work: Q4122344
cites work: Trigonometric series with general monotone coefficients
cites work: Trigonometric series of Nikol'skii classes
cites work: Weak type inequalities for moduli of smoothness: The case of limit value parameters
cites work: Ulyanov-type inequalities and generalized Liouville derivatives
cites work: Inequalities for moduli of smoothness versus embeddings of function spaces
cites work: THE IMBEDDING OF CERTAIN FUNCTION CLASSESHpω
cites work: Q4797357
DOI: 10.1007/S10476-017-0308-0
description: scientific article; zbMATH DE number 6837907