2009/09/23 by Erik Talvila, Talvila, Erik
Mathematics · #26A39 #42A85 #46E15 #46F10 #46G12 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #math.CA #math.FA #msc:26A39 #msc:42A85 #msc:46E15 #msc:46F10 #msc:46G12
paper · pdf · doi:10.48550/arxiv.0909.4336
arxiv created 2009/09/23 · arxiv updated 2009/12/01
If F is a continuous function on the real line and f=F' is its distributional derivative then the continuous primitive integral of distribution f is ∫abf=F(b)-F(a). This integral contains the Lebesgue, Henstock--Kurzweil and wide Denjoy integrals. Under the Alexiewicz norm the space of integrable distributions is a Banach space. We define the convolution f∗ g(x)=\intinf f(x-y)g(y) dy for f an integrable distribution and g a function of bounded variation or an L1 function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For g of bounded variation, f∗ g is uniformly continuous and we have the estimate ‖f∗ g‖_∞≤ ‖f‖‖g‖_\bv where ‖f‖=supI|∫If| is the Alexiewicz norm. This supremum is taken over all intervals I⊂\R. When g∈ L1 the estimate is ‖f∗ g‖≤ ‖f‖‖g‖1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.