I have trouble understanding the following: Let us assume that we have some $\Omega()$ data generating process, which generates {$h,x,y,z$} points. Let us assume that it is possible to construct the functional relationship of the form $h=f(x,y,z)$ by applying trilinear spline interpolation technique.
Now instead of $\Omega(h,x,y,z)$ generating process let's take into consideration the following process $\Omega(h,x,y^*,z^*)$, where $y^*$ and $z^*$ are scalars. Therefore we can construct $h=f(x)$ function using simple linear spline interpolation (for given values of $y,z$).
My question is the following: If we consider the following function $h=f(x,y^*,z^*)$ obtained from trilinear interpolation, does it coincide with $h=f(x)$ (for fixed $y^*,z^*$) obtained from linear spline interpolation?