On the convergence of Fourier series

Geraldo Soares de Souza

International Journal of Mathematics and Mathematical Sciences, 1984, vol. 7, 1-4

Abstract:

We define the space B p = { f : ( − π , π ] → R , f ( t ) = ∑ n = 0 ∞ c n b n ( t ) , ∑ n = 0 ∞ | c n | < ∞ } . Each b n is a special p -atom, that is, a real valued function, defined on ( − π , π ] , which is either b ( t ) = 1 / 2 π or b ( t ) = − 1 | I | 1 / p X R ( t ) + 1 | I | 1 / p X L ( t ) , where I is an interval in ( − π , π ] , L is the left half of I and R is the right half. | I | denotes the length of I and X E the characteristic function of E . B p is endowed with the norm ‖ f ‖ B p = Int ∑ n = 0 ∞ | c n | , where the infimum is taken over all possible representations of f . B p is a Banach space for 1 / 2 < p < ∞ . B p is continuously contained in L p for 1 ≤ p < ∞ , but different. We have THEOREM. Let 1 < p < ∞ . If f ∈ B p then the maximal operator T f ( x ) = sup n | S n ( f , x ) | maps B p into the Lorentz space L ( p , 1 ) boundedly, where S n ( f , x ) is the n t h -sum of the Fourier Series of f .

Date: 1984
References: Add references at CitEc
Citations:

Downloads: (external link)
http://downloads.hindawi.com/journals/IJMMS/7/895418.pdf (application/pdf)
http://downloads.hindawi.com/journals/IJMMS/7/895418.xml (text/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:895418

DOI: 10.1155/S0161171284000843

Access Statistics for this article

More articles in International Journal of Mathematics and Mathematical Sciences from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().