This allows one to fit an interpolated curve, where the y-coordinates of the curve must lie in a certain range (e.g. nowhere must the curve be below 0 or above 1 for any value of x), using any base interpolator and a function (and its inverse) that maps the constrained range (e.g. 0 to 1 inclusive) to -infinity to +infinity.
Let y(x) be an interpolated value and y_i the set of node values. An interpolated value can be written as y(x) = I(x, (x_i,y_i)) - for a given interpolator, it is a function of x, and the node coordinates (x_i,y_i). However even if all the node values (y_i) are constrained to the required range, there is no guarantee that y(x) will be in the range for all x (it depends on the interpolator).
Now let y*(x) and y*_i be the corresponding transformed values (i.e. the range is the entire real line), so y* = f(y). We may wish to work directly with the transformed values y*_i (since this allows unconstrained optimisation to find the values, where the interpolated curve is part of some larger calibration). To this end we have y(x) = f^-1[I(x,(x_i,y*_i))], which is not strictly an interpolator since the curve y(x) does not go through the points y*_i. We could of course write y(x) = f^-1[I(x,(x_i,f(y_i)))], which is a true interpolator (and could be write as I*(x,(x_i,y_i)) ). In both these cases y(x) is guaranteed to be in the range regardless of the base interpolator used.