Absztrakt: | The question of the strong approximation of functions $f$ belonging
to the space $L^p(R)$, $1 < p < \infty$, by its Dirichlet integral
$s_\nu (f,x) = {1\over \pi} \int_R f(x - t)\ {\sin \nu t\over
t} dt $ is studied. With the help of quotient $$d_\nu (f,p):=
\Biggl|\biggl\{ {1\over \nu} \int^\nu_0 |s_\mu(f,\cdot) - f(\cdot)|^p
d\mu \biggr\}^{1/p} \Biggr|_\infty, \quad \nu \in R_+,$$ the
class ${\cal L}_p(R) := \{f \in L^p(R): d_\nu(f,p) = O(\nu^{-1/p})$
as $\nu \to \infty\}$ is introduced. Sufficient conditions for
$f$ to belong to the class ${\cal L}_p(R)$ in the case $2 \leq
p <\infty$, and necessary conditions for $f$ to belong to ${\cal
L}_p(R)$ in the case $1 < p \leq 2$ are established. In the case,
when the Dirichlet integral is replaced by the Riesz means similar
results are obtained. Besides the problem of strong approximation
of a periodic function by the partial sum or Fej\'er mean of
its Fourier series is studied.}
\RV{D.K.Ugulawa (Tbilisi) |