This article is cited in
1 paper
On the upper bounds of Lebesgue constants for Forier–Jacobi series summation methods
O. L. Vinogradov St. Petersburg State University, Department of Mathematics and Mechanics
Abstract:
In what follows. the
$P^{(\alpha,\beta)}_k$ are Jacobi polinomians,
$C[a,b]$ is the space of continuous functions on
$[a,b]$ with uniform norm,
$\mathscr U^{\Lambda}_n\colon C[-1,1]\to C[-1,1]$ is a sequence of operators determined by a matrixof multipliers
$\Lambda=\{\lambda^{(n)}_k\}$:
\begin{gather*}
f\sim\sum^{\infty}_{k=0}a_kP^{(\alpha,\beta)}_k, \qquad \mathscr U^{\Lambda}_nf\sim\sum^{\infty}_{k=0}\lambda^{(n)}_ka_kP^{(\alpha,\beta)},
\\
\mathfrak L^{(\alpha,\beta)}_n(\Lambda)=\sup_{y\in[-1,1]}\sup_{\|f\|\le1}\left|\mathscr U^{\Lambda}_nf(y)\right|.
\end{gather*}
The values of $\sup\limits\mathfrak L^{(\alpha,\beta)}_n(\Lambda)$ and $\lim\limits_{n\to\infty}\mathfrak L^{(\alpha,\beta)}_n(\Lambda)$ are studied. It is proved that in the cases of
\begin{gather*}
1)\alpha=\beta=-1/2, \quad \lambda^{(n)}_k=\varphi(k/n);
\\
2)\alpha=\beta=1/2, \quad \lambda^{(n)}_k=\varphi((k+1)/n);
\\
3)\alpha=\beta=\pm1/2, \quad \lambda^{(n)}_k=\varphi((k+1/2)/n)
\end{gather*}
these values are equal to
$$
1) \quad \frac2\pi\int\limits^{\infty}_0\left|\int\limits^{\infty}_0\varphi(t)\cos zt\,dt\right|dz; \qquad 2,\ 3)\quad \frac2\pi\int\limits^{\infty}_0z\left|\int\limits^{\infty}_0t\varphi(t)\sin zt\,dt\right|dz.
$$
under some conditions on
$\varphi$.
Then it is shown that for the Legendre polynomials
$(\alpha=\beta=0)$ and
$\lambda^{(n)}_k=\varphi(k/n)$ the limit and the supremum of the Lebesgue constants may fail to be equal.
UDC:
517.5
Received: 14.06.2001