THE TEST CASE

We begin with a rough one-dimensional function. A random walk would be nice, the integral of random numbers (possibly coin flips). Call it r(x). Actually, I'd like a random walk that crosses the zero axis a couple times. We could try several seeds until we find an "attractive" one. Maybe leaky integrate random numbers.

Next, flex a piece of paper so that along the x-axis it matches r(x). Now any line parallel to the x-axis should match r(x). Let us tilt this on the y-axis, so our altitude function is h(x,y)= r(x) + y. Notice that we have a function whose second y-derivative vanishes everywhere. So $\delta_{yy}$ is a PEF for it. Its Gaussian curvature h_xxh_yy-h_xy² vanishes. Now, to confuse people, we rotate it $45^\circ$.Thus, I propose the test function h(x,y)= r(x+y) + (x-y). Its perfect PEF is

                .  0  0
                .  0  0
                1  0  0
                0 -2  0
                0  0  1

and we need to show that we can find it.

What will we give for data? I offer a track of dense data along the x axis (like a well log or seismogram). Already you can see with this and with the PEF, and one more data point off the track, the data everywhere is known. Well, almost everywhere. Actually, there is a shadow. Now we sprinkle a dozen data values around the plane. We might choose those points aliased on our rough function in such a way that everyone but us would be completely confused.