On the interrelation of the generalized Besov-Nikols'kiǐ and Weyl-Nikols'kiǐ classes of functions
type: Item
instance of: scholarly article
title: On the interrelation of the generalized Besov-Nikols'kiǐ and Weyl-Nikols'kiǐ classes of functions
zbMATH Open document ID: 0892.46029
DOI: 10.1007/BF01904789
author: B. V. Simonov
author: M. K. Potapov
published in: Analysis Mathematica
publication date: 1998-03-26T00:00:00Z
review text: The authors generalize former results of B. V. Simonov on embedding theorems for the Besov-Nikol'skij class and the Weyl-Nikol'skij class by considering spaces with more general weights in \(t\). The functions considered are \(2\pi\) periodic, measurable functions. The Besov-Nikol'skij space is defined by finiteness of the quantity \[ \int_0^1 \left(\omega_k(f,t)_p/t^r \right)^{\theta} \frac{dt}t, \] where \[ \omega_{\beta}(f,t)_p = \sup_{| h| \leq t} \Bigl\| \sum_{\nu = 0}^{\infty} (-1)^{\nu} \binom {\beta}{\nu} f(x + (\beta - \nu)h)\Bigr\| _p, \] with \[ \binom{\beta}{\nu} = \frac {\beta(\beta -1)\cdots (\beta - \nu +1)}{\nu !} \] and \(k>r>0\). The authors consider spaces in which the function \(\alpha(t) = t^{-r\theta -1}\) is replaced by a function for which there is a real number \(\sigma\) such that \[ \int _0^1 \alpha(t) t^{\sigma - \varepsilon} dt = +\infty,\quad \forall \varepsilon>0, \] while \[ \int _0^1 \alpha(t) t^{\sigma} dt < +\infty. \] The generalized Besov-Nikol'skiĭ class \(B(p, \theta, k, \alpha)\) is the set of \(f \in L_p\) for which \[ \int_0^1 \alpha(t) \omega_k^{\theta}(f,t)_p dt \] is finite. A function \(f\) belongs to the Weyl-Nikol'skij class if \(f \in L_p\) and its Fourier series \[ \frac{a_0(f)}2 + \sum_{n=1}^{\infty} (a_n(f) \cos nx + b_n(f) \sin nx), \] after multiplication by \(\lambda_n\), \[ \sum_{n=1}^{\infty} \lambda_n(a_n(f) \cos nx + b_n(f) \sin nx), \] where \[ \lambda_n = \left( \int_{1/n}^1 \alpha(t) dt + n^{k\theta} \int_0^{1/n} \alpha(t) t^{k\theta} dt \right)^{1/\theta}, \] is the Fourier series of some \(\phi \in L_p\). In the case of \(\alpha(t) = t^{-r\theta -1}\), \(\lambda_n = n^r\), and \(\phi\) is a fractional derivative of \(f\). The embedding theorems involve the spaces \(WH(\alpha, \beta, \psi, p, \theta, k)\) and \(B_0H(\alpha, \beta, \psi, p, \theta, k)\), where \(f \in WH\) if \(f \in W(p, \theta, k, \alpha)\) and \[ \omega_{\beta}(f, \delta)_p \leq C \psi(\delta), \quad 0< \delta \leq 1, \] and \(f \in B_…
reviewed by: Raymond Johnson
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 42A45
Mathematics Subject Classification ID: 42A55
zbMATH DE Number: 1134604
zbMATH Keywords: Besov-Nikols'kij space
zbMATH Keywords: Weyl-Nikols'kij space
zbMATH Keywords: embedding theorems
MaRDI profile type: Publication
cites work: Theorems on Fourier Series and Power Series
cites work: Q5774852
cites work: On \(\mathrm\text{ Ш}\)-rigidity of groups of order \(p^6\).
full work available at URL: https://doi.org/10.1007/bf01904789
OpenAlex ID: W1861258841
description: scientific article; zbMATH DE number 1134604