Looking for algebras designed to transform time series I am looking for information on (formal) algebraic systems that can be used to transform time-series - in either a practical or academic context.

I hope that there exists (at least one) small, expressive, set of operators - ranging over (finite) time-series.  I want to compare and contrast different systems  with respect to algebraic completeness, and brevity of representation, of common time-series transformations in various domains.

I realise this question is broad - but hope it is not too vague for datascience.stackexchange.  I welcome any pointers to relevant literature for specific scenarios, or the general subject.

Edit... (Attempt to better explain what I meant by an algebraic system...)

I was thinking about "abstract algebras" as discussed in Wikipedia:

http://en.wikipedia.org/wiki/Algebra#Abstract_algebra
http://en.wikipedia.org/wiki/Abstract_algebra#Basic_concepts

Boolean Algebras are (very simple) algebras that range over Boolean values.  A simple example of such an algebra would consist the values True and False and the operations AND, OR and NOT. One might argue this algebra is 'complete' as, from these two constants (free-variables) and three basic operations, arbitrary boolean functions can be constructed/described.

I am interested to discover algebras where the values are (time-domain) time-series.  I'd like it to be possible to construct "arbitrary" functions, that map time-series to time-series, from a few operations which, individually, map time-series to time-series.  I am open to liberal interpretations of "arbitrary". I would be especially interested in examples of these algebras where the operations consist 'higher-order functions' - where such operations have been developed for a specific domain.
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.