Fourier series of additive vector measures and their term-by-term differentiation

A. G. Areshkina

Matematičeskie zametki, Tome 64 (1998) no. 2, pp. 180-184

Résumé

On a measurable space $(T,\Sigma,\mu)$ we choose an additive measure $\nu\colon\Sigma\to Z$ ($Z$ is a Banach space) with the following property: for all $e\in\Sigma$, we have $\int _exd\nu=0\implies x\overset{\mu}{\sim} 0$; this measure defines an indefinite integral over the measure $\nu$ on $L^2(T,\Sigma,\mu)$. We prove that if $\{\tau_n(t)\}_{n=1}^\infty$ is an orthonormal basis in $L^2$ and $\theta _n(e)=\int_e\tau_n(t)d\nu$, then any additive measure $\nu\colon\Sigma\to Z$ whose Radon–Nikodým derivative $d\varphi/d\nu$ belongs to $L^2$ is uniquely expandable in a series $\varphi(e)=\sum_{n=1}^\infty\alpha_n\theta_n(e)$ that converges to $\varphi(e)$ uniformly with respect to $e\in\Sigma$ can be differentiated term-by-term, and satisfies $\sum_{n=1}^\infty\alpha_n^2\infty$. In the case $L^2[0,2\pi]$, $Z=\mathbb R$, the Fourier series of a $2\pi$-periodic absolutely continuous function $F(t)$ such that $F'(t)\in L^2[0,2\pi]$, is superuniformly convergent to $F(t)$.