Semi-periodic Interpolation

Question

I have a set of data, e.g. damped waves in a finite box of size L, so that the data is periodic in space but not time. At a particular instant in time, i.e. taking a cross section of data at a particular time, which I'll call dataSlice, I can create an interpolating function

intFn = Interpolation[dataSlice, InterpolationOrder -> 1, PeriodicInterpolation -> True]

and I don't run into trouble. However, if I attempt a periodic interpolation on the whole data set, it fails

intFn = Interpolation[dataSlice, InterpolationOrder -> 1, PeriodicInterpolation -> True]

(*In dimension 1 the data at the endpoints of the fundamental period are not equal*)

That makes sense, of course. For my data, it would make sense to do a semi-periodic interpolation. Does such a function exist? I would like it to function like

intFn = Interpolation[dataSlice, InterpolationOrder -> 1, PeriodicInterpolation -> {False,True,True,True}]

so the first (time) dimension would be non-periodic, and the spatial dimension is periodic.

Answer 1

If the data are equidistant in the two dimensions and if the InterpolationOrder is 1, BSplineFunction can be used equivalently.
The advantage is that BSplineFunctionis able to manage partially periodic domain.

n = 10;
dataSlice = N[Sin /@ Range[0, 2 Pi (1 - 1/n), 2 Pi/n] ]
data = Table[Exp[-2 i]  dataSlice, {i, 0.2, 1, 0.1}];
f = BSplineFunction[data, 2, SplineDegree -> 1, 
  SplineClosed -> {False, True}]
Plot3D[f[t, x], {x, 0, 2}, {t, 0, 2}]  

The domain of the basic pattern is [0,1]X[0,1] and the samples are equidistant in both dimensions of this domain.

I don't know if it works for Interpolation order >1 (I would guess : No).