The limit of the Lebesgue constants of summation methods of Fourier–Legendre series determined by a multiplier function

In what follows, $C[-1,1]$ is the space of continuous functions $f\colon[-1,1]\to\mathbb R$ with uniform norm, $P_k$ are the Legendre polynomials such that $P_k(1)=1$, $J_0$ is the Bessel function of zero index. We consider sequences of linear operators (summation methods) $U_n\colon C[-1,1]\to C[-1,1]$ determined by a multiplier function $\varphi$:
$$ U_nf(y)=\int\limits_{-1}^1f(x)\sum_{k=0}^{\infty}\varphi(k/n)(k+1/2)P_k(y)P_k(x)\,dx. $$
The norms $\mathfrak L_n$ of the operators $U_n$ are called the Lebesgue constants of the summation method. The main result is the following.
If $\varphi$ is continuous on $[0,+\infty)$,
\begin{gather*} \sum_{k=0}^{\infty}\varphi^2(k/n)(k+1/2)<\infty \text{ for each </nomathmode><mathmode>$n\in\mathbb N$,} \qquad \int\limits_0^\infty\varphi^2(x)x dx<\infty;
B\varphi(z)=z\int\limits_0^\infty\varphi(x)xJ_0(zx) dx \end{gather*}
</mathmode><nomathmode> is the Fourier–Bessel transformation of $\varphi$, and the function $z^{q-1}|B\varphi(z)|^q$ is summable on $[0,+\infty)$ with some $q>1$, then
$$ \lim_{n\to\infty}\mathfrak L_n=\int\limits_0^\infty|B\varphi|. $$