Sequence operator and transform

In summary, a shift polynomial sequence operator, denoted as S, is defined as the sum from i=0 to m of c_i times the shift operator E raised to the power of i. When applied to a sequence a_n, S(a)_n is equal to the sum from i=0 to m of c_iE^ia_n. When S is composed with itself k times, it is defined as S^k(a)_n = [sum from i=0 to m of c_iE^i]^ka_n. If the first elements from each new sequence for each k are taken, it is defined as S^n(a)_0 = b_n = T(a)_n. The goal is to find a general formula
  • #1
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Define a shift polynomial sequence operator as
[tex]S=\sum_{i=0}^mc_i\mathbf E^i[/tex]

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

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

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

I want to find a general formula for [itex]T^k(a)_n=b_n[/itex] and it's inverse, [itex]T^{-k}(b)_n=a_n[/itex] where [itex]S[/itex] is any shift polynomial operator. Any help would be greatly appreciated.Thanks
bq
 
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  • #2
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##.
 

1. What is a sequence operator?

A sequence operator, also known as a sequence function, is a mathematical function that takes a sequence of values as input and produces a single value as output. It is used to manipulate and transform sequences of data.

2. What are some common sequence operators?

Some common sequence operators include map, filter, reduce, take, skip, and concat. These operators are used to perform operations such as transforming, filtering, and combining elements in a sequence.

3. How do sequence operators differ from regular functions?

Sequence operators differ from regular functions in that they operate on sequences of data rather than individual values. They also often use lazy evaluation, meaning that the output is only computed when it is needed, rather than all at once.

4. What is the purpose of a transform function?

A transform function is used to transform one sequence into another sequence. It takes a sequence as input, applies a transformation to each element in the sequence, and outputs a new sequence with the transformed elements.

5. How can sequence operators and transforms be useful in data analysis?

Sequence operators and transforms are essential tools in data analysis as they allow for efficient manipulation and transformation of large datasets. They can be used to perform operations such as cleaning and organizing data, extracting specific information, and creating new datasets for further analysis.

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