A generalization of Volkenborn integral

The purpose of this thesis is to extend the Volkenborn integral of a function by replacing the Haar distribution with weakly Volkenborn distributions. We prove that all functions f ∈ C1($Zp,Cp$) are Volkenborn integrable with respect to weakly Volkenborn distributions and find relations between the Volkenborn integral of a C1 function and its Mahler coefficients. We show how the log circular unit distribution can define an analytic function which interpolates p-adiclly the L-values L′ (χ, 0), where L is the complex L-function. Then we define $fm$, the Radon-Nikodym derivative of a weakly Volkenborn distribution $m$, with respect to Haar distribution and show that Radon-Nikodym derivatives of strongly Volkenborn distributions are Lipshitz functions. Furthermore, for any C 1 function f, we associate a strongly Volkenborn distribution $mf$, and prove that $fmf$ = f. Finally if $m$ is a strongly Volkenborn distribution $m$, with a C1 Radon-Nikodym derivative $fm$, then $m=mfm+m1$, where $m1$ is a measure on $Zp$. This last identity resembles the Lebesgue decomposition of a complex measure into an absolute continuous measure and a singular one.