We present here a vector calculus on weighted networks following the guidelines of Differential Geometry. The key to develop an efficient calculus on weighted networks which mimetizes the calculus in the smooth case is an adequate construction of the tangent space at each vertex. This allows to consider discrete vector fields, inner products and general metrics. Then, we obtain discrete version...

We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted l2 norms of the Dirichlet coefficients. We characterize the multiplier algebras for some of these spaces. 0 Introduction Let w = {wn}n=n0 be a sequence of positive numbers. In this paper we are concerned with Hilbert spaces of functions representable by Dirichlet series: H w = {

For a finite positive Borel measure μ on the unit circle, let D(μ) be associated harmonically weighted Dirichlet space. A shift invariant subspace M recognizes strong approximate spectral cosynthesis if there exists sequence of subspaces M_k, with codimension, such that orthogonal projections onto M_k converge in operator topology to projection M. If is sum atoms, then we show admits cosynthesis.

In this paper we prove an existence and uniqueness result for the double phase Dirichlet problem when lowest exponent is equal to 1. Our solution a function of bounded variation that simultaneously lies in suitable weighted Sobolev space found as limit sequence solutions intermediate problems whose $p$ goes As that, our approach involves study some relevant properties generalized Orlicz-Sobolev...

D |f ′|2 dx dy. In particular, we will cover the basic structure of these functions their boundary values and their zeros along with two important operators that act on this space of functions the forward and backward shifts. This survey is by no means complete. For example, we will not cover the Toeplitz or Hankel operators on these functions, nor will we cover the important topic of interpola...