Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 163, January 1972
CONVERGENCE, UNIQUENESS, AND SUMMABILITY OF
MULTIPLE TRIGONOMETRIC SERIES
BY
J. MARSHALL ASH(1) AND GRANT V. WELLAND(2)
Abstract. In this paper our primaryinterestis in developingfurtherinsight into
convergencepropertiesof multipletrigonometricseries,withemphasison theproblem
of uniquenessof trigonometricseries.LetE be a subsetof positive(Lebesgue)measure
of the k dimensionaltorus. The principalresultis that the convergenceof a trigono-
metricserieson E forces the boundednessof the partialsums almost everywhereon
Ewherethesystemofpartialsumsistheoneassociatedwiththesystemof allrectangles
situatedsymmetricallyabout the origin in the lattice plane with sides parallelto the
axes. If E has a countablecomplement,then the partialsums are bounded at every
point of E. This resultimpliesa uniquenesstheoremfor double trigonometricseries,
namely,thatif a doubletrigonometricseriesconvergesunrestrictedlyrectangularlyto
zero everywhere,then all the coefficientsare zero. Although uniquenessis still con-
jecturalfordimensionsgreaterthantwo, we obtainpartialresultsandindicatepossible
lines of attackfor this problem.

  

Source: Ash, J. Marshall - Department of Mathematical Sciences, DePaul University

 

Collections: Mathematics