On the upper bounds of Lebesgue constants for Forier–Jacobi series summation methods

O. L. Vinogradov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 29, Tome 282 (2001), pp. 34-50

Résumé

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.