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Summability of Fourier transforms of functions from Lorentz spaces
Luleå University of Technology, Department of Engineering Sciences and Mathematics.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This PhD thesis is devoted to the study of relations between integrability properties of functions and summability properties of its Fourier coefficients and transforms. The relations are given in terms of generalized weighted Lorentz norms, where the weights have some additional growth properties. The thesis contains six papers (papers A-F) together with an introduction, which put these papers into a general frame.

In paper A some relations between weighted Lorentz norms and some corresponding sums of Fourier coefficients are studied 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 generalized weighted Lorentz spaces in the case of a regular system. Some new inequalities of Hardy-Littlewood-Paley type with respect to a regular system for these generalized Lorentz spaces are obtained. It is also proved that the obtained results are in a sense sharp.

In paper C we investigate integrability properties of the orthogonal series with coefficients from generalized weighted Lorentz spaces in the case of a regular system. The upper and the lower estimates of some corresponding Lorentz type norms of the Fourier coefficients are obtained.

In paper D some new Boas type theorems for generalized weighted Lorentz spaces with respect to regular systems for generalized monotone functions are proved.

In paper E inequalities for the Fourier transform of functions from the generalized weighted Lorentz spaces are studied. The upper and the lower estimates of the norm of the Fourier transform in generalized weighted Lorentz spaces are derived.

Finally, in paper F a new inequality concerning the Fourier transform is derived. Moreover, it is described conditions so that this result is sharp in the sense that both upper and lower bounds are obtained.

Place, publisher, year, edition, pages
Luleå: Luleå University of Technology, 2017.
Series
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-63150ISBN: 978-91-7583-898-4 (print)ISBN: 978-91-7583-899-1 (electronic)OAI: oai:DiVA.org:ltu-63150DiVA: diva2:1090966
Public defence
2017-06-08, E246, Luleå, 10:00 (English)
Opponent
Supervisors
Available from: 2017-04-25 Created: 2017-04-25 Last updated: 2017-05-18Bibliographically approved
List of papers
1. Some new results concerning the Fourier coefficients in Lorentz type spaces
Open this publication in new window or tab >>Some new results concerning the Fourier coefficients in Lorentz type spaces
2010 (English)Report (Other academic)
Place, publisher, year, edition, pages
Luleå: Department of Mathematics, Luleå University of Technology, 2010. 15 p.
Series
Gula serien, ISSN 1400-4003 ; 2010:05
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-23326 (URN)680c7c1f-46cb-4c28-81e7-a184aac72f36 (Local ID)680c7c1f-46cb-4c28-81e7-a184aac72f36 (Archive number)680c7c1f-46cb-4c28-81e7-a184aac72f36 (OAI)
Note
Godkänd; 2010; 20120207 (andbra)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-04-25Bibliographically approved
2. Some new two-sided inequalities concerning the Fourier transform
Open this publication in new window or tab >>Some new two-sided inequalities concerning the Fourier transform
2017 (English)In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 20, no 3, 855-864 p.Article in journal (Refereed) Published
Abstract [en]

The classical Hausdorff-Young and Hardy-Littlewood-Stein inequalities do not hold for p > 2. In this paper we prove that if we restrict to net spaces we can even derive a two-sided estimate for all p > 1. In particular, this result generalizes a recent result by Liflyand E. and Tikhonov S. [7] (MR 2464253).

Place, publisher, year, edition, pages
Element D.O.O., 2017
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-60120 (URN)10.7153/mia-20-54 (DOI)2-s2.0-85020774620 (Scopus ID)
Note

Validerad;2017;Nivå 2;2017-07-03 (andbra)

Available from: 2016-11-03 Created: 2016-11-03 Last updated: 2017-07-07Bibliographically approved
3. Relations between summability of the Fourier coefficients in regular systems and functions from some Lorentz type spaces
Open this publication in new window or tab >>Relations between summability of the Fourier coefficients in regular systems and functions from some Lorentz type spaces
2010 (English)In: Proceedings of A. Razmadze Mathematical Institute, ISSN 1512-0007, Vol. 152, 73-88 p.Article in journal (Refereed) Published
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-2403 (URN)005b2f80-0757-11e0-b767-000ea68e967b (Local ID)005b2f80-0757-11e0-b767-000ea68e967b (Archive number)005b2f80-0757-11e0-b767-000ea68e967b (OAI)
Note
Validerad; 2010; 20101214 (andbra)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-04-25Bibliographically approved
4. On summability of the Fourier coefficients in bounded orthonormal systems for functions from some Lorentz type spaces
Open this publication in new window or tab >>On summability of the Fourier coefficients in bounded orthonormal systems for functions from some Lorentz type spaces
2010 (English)In: Eurasian Mathematical Journal, ISSN 2077-9879, Vol. 1, no 2, 76-85 p.Article in journal (Refereed) Published
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-7127 (URN)571c3de0-0758-11e0-b767-000ea68e967b (Local ID)571c3de0-0758-11e0-b767-000ea68e967b (Archive number)571c3de0-0758-11e0-b767-000ea68e967b (OAI)
Note
Godkänd; 2010; 20101214 (andbra)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2017-04-25Bibliographically approved

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