Two examples concerning almost continuous functions

On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

We comment on a Mazur problem from "Scottish Book" concerning second partial derivatives. It is proved that, if a function $f(x,y)$ of real variables defined on a rectangle has continuous derivative with respect to $y$ and for almost all $y$ the function $\,F_y(x):=f'_y(x,y)$ has finite variation, then almost everywhere on the rectangle there exists the partial derivative $f"_{yx}$. We construct a separately twice differentiable function, whose partial derivative $f'_x$ is discontinuous with respect to the second variable on a set of positive measure. This solves in the negative the Mazur prob…

Factorization of Scalar Piecewise Continuous Almost Periodic Functions

This paper is to characterize piecewise continuous almost periodic functions as the product of Bohr almost periodic functions and sequences. As an application, the result is used to discuss piecewise continuous almost periodic solutions of impulsive differential equations.

Almost everywhere convergence of convolution powers without finite second moment

We generalize a theorem of Bellow and Calderón concerning the a.e. convergence of the convolution powers $\ds μ^nf(x)=\sum_{k}μ^n(k)f(T^k x)$ where $T$ is a measure preserving transformation of a probability space and $μ$ is a probability measure on the integers.

Almost everywhere convergence of Fourier series on SU(2): the case of Holder continuous functions

We consider an aspect of the open problem: Does every square-integrable function on SU(2) have an almost everywhere convergent Fourier series? Let 0 < alpha < 1. We show that to each countable set E in SU(2) there corresponds an alpha-Holder continuous function on SU(2) whose Fourier series diverges on E. We also show that the Fourier series of each alpha-Holder continuous function on SU(2) converges almost everywhere.

向量值分段连续几乎自守函数及其若干推论

Vector valued piecewise continuous almost automorphic functions and some consequences

本文将分段连续 $\mathbb{Z}$-几乎自守函数推广到 Banach 空间，建立其代数性质与紧几乎自守刻画，并应用于序列、微分方程及热方程。

Almost everywhere convergence questions of series of translates of non-negative functions

This survey paper is based on a talk given at the 44th Summer Symposium in Real Analysis in Paris. This line of research was initiated by a question of Haight and Weizsäker concerning almost everywhere convergence properties of series of the form $\sum_{n=1}^{\infty}f(nx)$. A more general, additive version of this problem is the following: Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is of type 1 if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {\mathbb R}\to [0,+ {\infty})$ either the …