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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:$$\label{f--}c_1\left\|\overline{f}\right\|_{L_p\left[0,1\right]}^p\leq \sum_{k=1}^{\infty}k^{p-2}|a_k|^{p}\leqc_2\left\|tf'\right\|_{L_p\left[0,1\right]}^p,\;\;\;\text{for}\;1$$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.
##### Series
Licentiate thesis / Luleå University of Technology, ISSN 1402-1757
##### Keyword [en]
Lorentz spaces, Fourier series, Inequalities, Mathematics
Matematik
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
Local ID: 0042b410-e65e-11df-8b36-000ea68e967bISBN: 978-91-7439-170-1 (print)OAI: oai:DiVA.org:ltu-16787DiVA: diva2:989774
##### Note
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: 2016-09-29 Last updated: 2017-11-24Bibliographically approved

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Cite
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