[ceeollrz] Continuous population density function

Given aggregate census information of a country partitioned into regions and the population of each region, construct a population density total function giving a density at any point.

A simple way to do this is to evenly distribute a region's population over the area.  This might be enough for many purposes.

There are discontinuities at the boundaries.  Construct a function that avoids such discontinuities, perhaps a union of plane segments.  Construct a function the avoids discontinuities of the first derivative.  Essentially, we want spline surfaces, though over weird areas and not defined the usual way with control points.  I suspect the answer is a system of integro-differential equations to be solved numerically.

(Though in real life, discontinuity at a boundary might be fine if it represents a difficult to traverse physical boundary, perhaps a river.)

Such an artificially constructed surface is necessarily a lie.  Provide another function that gives the uncertainty of the density function which can be used in further analyses:  For example, suppose we wish to sum the population of only a portion of a region.  Integrate the constructed surface over the area.  However, the answer very uncertain.  We do not know what the population density inside the region.