Change search
ReferencesLink to record
Permanent link

Direct link
Relations between functions from some Lorentz type spaces and summability of their Fourier coefficients
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
2010 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This Licentiate Thesis is devoted to the study of summability of the Fourier coefficients for functions from some Lorentz type spaces and contains three papers (papers A - C) together with an introduction, which put these papers into a general frame.Let $\Lambda_p(\omega),\;\; p>0,$ denote the Lorentz spaces equipped with the (quasi) norm$$\|f\|_{\Lambda_p(\omega)}:=\left(\int_0^1\left(f^*(t)\omega(t)\right)^p\frac{dt}{t}\right)^{\frac1p}$$for a function $f$ on [0,1] and with $\omega$ positive and equipped with some additional growth properties.In paper A some relations between this quantity and some corresponding sums of Fourier coefficients are proved for the case with a general orthonormal bounded system. Under certain circumstances even two-sided estimates are obtained.In paper B we study relations between summability of Fourier coefficients and integrability of the corresponding functions for the generalized spaces $\Lambda_p(\omega)$ in the case of a regular system. For example, all trigonometrical systems, the Walsh system and Prise's system are special cases of regular systems. Some new inequalities of Hardy-Littlewood-P\'{o}lya type with respect to a regular system for the generalized Lorentz spaces $\Lambda_p(\omega)$ are obtained. It is also proved that the obtained results are in a sense sharp.The following inequalities are well-known:\begin{equation}\label{f--}c_1\left\|\overline{f}\right\|_{L_p\left[0,1\right]}^p\leq \sum_{k=1}^{\infty}k^{p-2}|a_k|^{p}\leqc_2\left\|tf'\right\|_{L_p\left[0,1\right]}^p,\;\;\;\text{for}\;1\end{equation}where $\overline{f(t)}=\frac1t\left|\int_0^tf(s)ds\right|$ and $f'(t)$ is the derivative of the function $f(t).$ (Here $\{a_k\}_{k=1}^\infty $ are the Fourier coefficients of the function $f$). In paper C we prove some analogues Hardy-Littlewood-P\'{o}lya type inequalities \eqref{f--} with respect to the regular system for generalized Lorentz spaces $\Lambda_{p}(\omega).$

Place, publisher, year, edition, pages
Luleå: Luleå tekniska universitet, 2010.
Licentiate thesis / Luleå University of Technology, ISSN 1402-1757
Keyword [en]
Lorentz spaces, Fourier series, Inequalities, Mathematics
Keyword [sv]
Research subject
URN: urn:nbn:se:ltu:diva-16787Local ID: 0042b410-e65e-11df-8b36-000ea68e967bISBN: 978-91-7439-170-1OAI: diva2:989774
Godkänd; 2010; 20101102 (aigkop); LICENTIATSEMINARIUM Ämnesområde: Matematik/Mathematics Examinator: Professor Lars-Erik Persson, Luleå tekniska universitet Diskutant: Docent Natasha Samko, Universidade do Algarve, Portugal Tid: Måndag den 20 december 2010 kl 10.15 Plats: D2214-15, Luleå tekniska universitetAvailable from: 2016-09-29 Created: 2016-09-29Bibliographically approved

Open Access in DiVA

fulltext(603 kB)0 downloads
File information
File name FULLTEXT01.pdfFile size 603 kBChecksum SHA-512
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Kopezhanova, Aigerim
By organisation
Mathematical Science

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

ReferencesLink to record
Permanent link

Direct link