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

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.