Looking for algebras designed to transform time series
It seems like you are overthinking this and getting caught up in your own head a little.

We can define a restricted set of time series as those which are well behaved (non-asymptotic) and single valued.  These time series then form a time series algebra which have additive, multiplicative, commutative, and associative properties.  There is an identity element (straight line with value 1) and a zero element (straight line with value 0).  We even have a well defined inverse member for all time series not crossing zero.  I believe this even forms an Abelian group.

Further, previous posters are correct that this group can be perfectly represented by various orthogonal vector spaces such as polynomials and Fourier space.  This is true whether we are talking about Lagrange polynomials, Fourier series, or Gauss-Lobatto-Legendre polynomials.

The point then becomes... where does this get you? You got caught up in the suggestion to employ Fourier series in that you didn't want to remain in frequency space, but this is entirely unnecessary.  You can, for instance, Fourier transform into frequency space, apply a high frequency filter, and then transform back into the time domain.  This is incredibly effective in removing noise (although band pass filters tend to work better).

Stepping back from the question a little, it seems like you might really be asking:  Given typical time series and typical time series operations (smoothing, de-seasoning, de-trending, etc) do the time series and their typical operations constitute a group with a well defined algebra?  The answer is, yes.  All of these operations just involve finding other time series which can be added, subtracted, multiplied, or divided by the original time series in order to yield a more physically tractable (interpretable) representation.