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On upper estimates of the partial sums of a trigonometric series in terms of lower estimates
A. S. Belov
Abstract:
Let
$\{a_k\}_{k=0}^\infty$ and
$\{b_k\}_{k=0}^\infty$ be sequences of real numbers and let
$ S_n(x)$ be defined by
$$
S_n(x)=\sum^n_{k=0}\bigl(a_k\cos(kx)+b_k\sin(kx)\bigr),\qquad
n=0,1,\dotsc\,.
$$
It is proved that the estimate
$$
\max_x S_n(x)\leqslant 4a_0 n^{1-\alpha},
$$
holds for each natural number
$n$ such that
$S_m(x)\geqslant0$ for all
$x$ and
$m=1,\,\dots,\,n$. Here
$\alpha\in(0,\,1)$ is the unique root of the equation
$$
\int^{3\pi /2}_0 t^{-\alpha}\cos t\,dt=0.
$$
It is proved that the order
$n^{1-\alpha}$ in this estimate cannot be improved. Various generalizations of this result are also obtained.
UDC:
517.5
MSC: Primary
42A05; Secondary
42A32,
42B05 Received: 13.02.1992
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