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 $T$.