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Zapiski Nauchnykh Seminarov POMI, 2001, Volume 282, Pages 34–50
(Mi znsl1505)
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This article is cited in 1 scientific paper (total 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.
Received: 14.06.2001
Citation:
O. L. Vinogradov, “On the upper bounds of Lebesgue constants for Forier–Jacobi series summation methods”, Investigations on linear operators and function theory. Part 29, Zap. Nauchn. Sem. POMI, 282, POMI, St. Petersburg, 2001, 34–50; J. Math. Sci. (N. Y.), 120:5 (2004), 1662–1671
Linking options:
https://www.mathnet.ru/eng/znsl1505 https://www.mathnet.ru/eng/znsl/v282/p34
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