Sequence operator and transform

[bqllpd]

Define a shift polynomial sequence operator as
S=\sum_{i=0}^mc_i\mathbf E^i

where \mathbf E is the shift operator and c_i are some constants, variables, functions, etc. When S is applied to a sequence \{a_n\}, then S(a)_n=\sum_{i=0}^mc_i\mathbf E^ia_n.

If S is composed with itself k times with the sequence, then define
S^k(a)_n=\left[\sum_{i=0}^mc_i\mathbf E^i\right]^ka_n.

If the first elements from each new sequence for each k are taken, define this as S^n(a)_0=b_n=T(a)_n. This is a problem I'm having. I know what T^k(a)_n=b_n and T^{-k}(b)_n=a_n are when m=1, c_0=\pm k when c_1=1 and c_0=\pm 1 when c_1=-1.

I want to find a general formula for T^k(a)_n=b_n and it's inverse, T^{-k}(b)_n=a_n where S is any shift polynomial operator. Any help would be greatly appreciated.Thanks
bq

 

[fresh_42]

The definition $S(a)_n := S(a_n)$ looks a bit strange, as I would have expected $S(a)_n = (S(a))_n$ because shifts from other positions could end up at position $n$, whereas you defined it componentwise, i.e. $S(a)_n$ is itself a finite sequence.

Before you go ahead, this has to be clarified. Also a general rule for such equations is always to make some easy examples. E.g. start with $E(n)=n+1$ and $a_n=\frac{1}{n}$ and then write down a few more complicated examples. This should help you to define a precise definition for $S(a)$ as well as $T$.