ON THE COEFFICIENTS OF MULTIPLE WALSH-FOURIER SERIES WITH SMALL GAPS

Journal of Mathematical Sciences

Ghodadra,Bhikha Lila¹ 

关键词: Multiple Walsh-Fourier coe cient;    Function of bounded p-variation in several variables;    order of magnitude;    Lacunary Fourier series with small gaps.;   

DOI  :  

学科分类：数学（综合）

来源: University of Tokyo * Department of Mathematical Sciences

【 摘 要 】

ForaLebesgueintegrablecomplex-valuedfunction$f$definedoverthe$m$-dimensionaltorus$mathbb{I}^m:=[0,1)^m$,let$hatf({fn})$denotethemultipleWalsh-Fouriercoefficientof$f$,where${fn}=left(n^{(1)},dots,n^{(m)}ight)in(mathbb{Z}^+)^m$,$mathbb{Z^+}=mathbb{N}cup{0}$.TheRiemann-Lebesguelemmashowsthat$hatf({fn})=o(1)$as$|{fn}|oinfty$forany$fin{mL}^1(mathbbI^m)$.However,itisknownthat,theseFouriercoefficientscantendtozeroasslowlyaswewish.ThedefinitiveresultisduetoGhodadraBhikhaLilaforfunctionsofbounded$p$-variation.Weshallprovethatthisisjustamatteronlyoflocalbounded$p$-variationforfunctionswithmultipleWalsh-Fourierserieslacunarywithsmallgaps.Ourresults,asinthecaseoftrigonometricFourierseriesduetoJ.R.PatadiaandR.G.Vyas,illustratetheinterconnectionbetween`localness'ofthehypothesisand`typeoflacunarity'andallowustointerpolatetheresults.

【 授权许可】

Unknown