Norms of Linear Functionals, Summability of Trigonometric Fourier Series and Wiener Algebras

Abstract

Let $L_{0,1}(\mathbb {R}_+)$ be the Banach space endowed with the norm

$$\displaystyle \|f\|{ }_{0,1}=\int _0^\infty f^*,\qquad f^*(x)=\underset {t\geq x}{\text{ess sup}}~|f(t)|, $$

while $L_{0,\infty }(\mathbb {R}_+)$ with the norm $\displaystyle \sup \limits _{x>0}\frac {1}{x}\int _0^x|f|$.

First, the norms of continuous linear functionals are calculated in these spaces (Theorems 1 and 2). Secondly, these theorems are applied to the problems of summability of trigonometric Fourier series, as ε → 0, by the linear means of type

$$\displaystyle \sum \limits _{k\in \mathbb {Z}}\varphi (k\varepsilon )\widehat {f}_ke^{ikx}, $$

where $\widehat {f}_k$ are the Fourier coefficients of f, $k\in \mathbb {Z}$, according to the function $\varphi :\mathbb {R}\rightarrow \mathbb {C}$.

Theorem 6 is a criterion, that is, a necessary and sufficient condition for the summability at every Lebesgue point (almost everywhere).

In Theorem 3, a general sufficient condition for the boundedness in ε on a set wider than that of Lebesgue points of the norms of these means as functionals is obtained. As a consequence, a general sufficient condition for the convergence of the mentioned means as ε → 0 at all the points of differentiability of indefinite integral of f (d-points) is proved (Theorem 4), while for compactly supported φ a necessary condition is given.

The results are accompanied by examples.