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

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.