We study the differential properties of higher-order statistical probabilistic programs with recursion and conditioning. Our starting point is an open problem posed by Hongseok Yang: what class have densities that are differentiable almost everywhere? To formalise problem, we consider Statistical PCF (SPCF), extension call-by-value real numbers, constructs for sampling give SPCF a sampling-styl...

It is proved that the maximal operators of subsequences N?rlund logarithmic means and Ces?ro with varying parameters Walsh-Fourier series bounded from dyadic Hardy spaces Hp to Lp. This implies an almost everywhere convergence for summability means.

The aim of matrix recovery is to recover P ∈ M ⊂ F p × q from L A ( ) = Tr 1 T , 2 … N with j V which raised in many areas. In this paper, we build up a framework for almost everywhere means injectivity on . We mainly focus the following question: how measurements are needed all matrices ? For case where both and algebraic varieties, use tools geometry study question present some results addres...

In this paper we present some results on convergence and summability of oneand multi-dimensional trigonometric andWalsh-Fourier series. The Fejér and Cesàro summability methods are investigated. We will prove that the maximal operator of the summability means is bounded from the corresponding classical or martingale Hardy space Hp to Lp for some p > p0. For p = 1 we obtain a weak type inequalit...

Every one of the important strong limit theorems that we have seen thus far – the strong law of large numbers, the martingale convergence theorem, and the ergodic theorem – has relied in a crucial way on a maximal inequality. This is no accident: it can in fact be shown that a maximal inequality is a necessary condition for an almost everywhere convergence theorem. We will refrain from carrying...

Abstract. We establish two new characterizations of magnetic Sobolev spaces for Lipschitz magnetic fields in terms of nonlocal functionals. The first one is related to the BBM formula, due to Bourgain, Brezis, and Mironescu. The second one is related to the work of the first author on the classical Sobolev spaces. We also study the convergence almost everywhere and the convergence in L appearin...

In this paper we prove analogs of Korovkin’s theorem in the context weakly nonlinear and monotone operators acting on Banach lattices functions several variables. Our results concern convergence almost everywhere, measure $$L^{p}$$ -norm. Several illustrating theory are also included.