Uniform Lebesgue constants of local spline approximation

Let a function $\varphi\in C^1[-h,h]$ be such that $\varphi(0)=\varphi'(0)=0$, $\varphi(-x)=\varphi(x)$ for $x\in [0;h])$, and $\varphi(x)$ is nondecreasing on $[0;h]$. For any function $f:\ \mathbb R\to \mathbb R$, we consider local splines of the form
$$S(x)=S_{\varphi}(f,x)=\sum_{j\in \mathbb Z} y_j B_{\varphi}\Big( x+\frac{3h}{2}-jh\Big)\quad (x\in \mathbb R),$$
where $y_j=f(jh)$, $m(h)>0$, and
$$B_{\varphi}(x)=m(h)\left\{
\begin{array}{cl}\varphi(x),& x\in [0;h],\\ 2\varphi(h)-\varphi(x-h)-\varphi(2h-x),& x\in [h;2h], \\ \varphi(3h-x),& x\in [2h;3h],\\ 0, & x\not\in [0;3h]. \end{array}
\right.$$
These splines become parabolic, exponential, trigonometric, etc., under the corresponding choice of the function $\varphi$. We study the uniform Lebesgue constants $L_{\varphi}=\|S\|_C^C$ (the norms of linear operators from $C$ to $C$) of these splines as functions depending on $\varphi$ and $h$. In some cases, the constants are calculated exactly on the axis $\mathbb R$ and on a closed interval of the real line (under a certain choice of boundary conditions from the spline $S_{\varphi}(f,x)$).