Convergence and integrability for some classes of trigonometric series
Volume 420 / 2003
Abstract
\noindent
In this work, the
theory of $L^1$-convergence
for some classes of trigonometric series is elaborated. The work contains
four chapters in which some new results are obtained. Also, new proofs of
some well known theorems are given. A classical result concerning the
integrability and $L^1$-convergence of a cosine series
$${{a_0}\over{2}}+\sum_{n=1}^\infty a_n\cos nx\leqno{\rm (C)}$$
with convex coefficients is the well known theorem of Young.
Later, Kolmogorov
extended Young's result for series (C) with quasi-convex coefficients and
also showed that such cosine series converge in $L^1$-norm if and
only if $a_n\log n=o(1)$ as $n\to\infty$.
In 1973, S.~A.~Telyakovski\u\i extended the %old classical
result of Kolmogorov. Namely,
he redefined the class of numerical sequences introduced by S.~Sidon.
He denoted this class by $S$
and proved, first, that the Sidon class is equivalent to
$S$, and second, that $S$ is an
$L^1$-integrability class for series (C). Also, he
proved that if $\{a_n\}_{n=0}^\infty\in S$,
then the
series (C)
converges
in $L^1$-norm iff $a_n\log n=o(1)$, $n\to\infty$. The class $S$ is
usually called the Sidon–Telyakovski\u\i class. Several authors,
Boas, Fomin, \v{C}. Stanojević, Bojanić,
and others have extended
these
classical results by addressing
one or both of the following two questions:
\vskip4pt
{(i)} If $\{a_n\}$ belongs to the class
BV of null sequences of bounded variation,
is (C) the Fourier series of its sum~$f$?
{(ii)} If $\{a_n\}\in BV$, is (C) the
Fourier series of some function $f\in L^1$ and is it true
that $\Vert S_n-f\Vert=o(1)$
as $n\to\infty$ iff $a_n\log n=o(1)$, $n\to\infty$?
\vskip4pt
Here,
$S_n$ denotes the $n$th partial sum of the series (C), and
$\Vert\,\cdot\,\Vert$ is the $L^1$-norm. Fomin, Stanojević,
Tanović-Miller and others
have given a positive answer to question (ii)
for many
classes of sequences.
We consider the
problem of $L^1$-convergence of the $r$th
derivative of Fourier series, i.e. we define some new integrability
classes of
the $r$th derivative of Fourier series. Some necessary
and sufficient conditions for $L^1$-convergence of the
$r$th derivative of Fourier series are obtained.
We define some new $L^p$ $(0
