Absztrakt: | The authors give a new characterization of functions $f$ defined
on the real line $(-\infty, \infty)$ in order to belong to a
Besov space $B\sp{p, r}\sb \alpha$ for some $0< \alpha< 1$ and
$1\le p$, $r\le \infty$. These conditions are in terms of the
Riesz mean of $f$ in case $1\le p\le \infty$, and in terms of
the Dirichlet integral of $f$ in case $1< p< \infty$. An analogous
characterization of periodic functions on the torus $[-\pi, \pi)$
was initiated by Fourier and Self, via the partial sums of their
Fourier series. The novelty in the authors' treatment is that
they use norms involving integrals, instead of norms involving
sums of infinite series. Their approach is also appropriate to
building up a complete characterization of Besov spaces on the
torus. |