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In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer science), I have come upon the necessity of solving (or at least finding out some of the solution's properties) infinite systems of difference (NOT differential) equations (linear, autonomous, first-order). Solving countable systems is sufficient for my purpose.

However, I have almost no idea about how to solve such systems (and what are the solvability conditions). I have a feeling that a possible approach is to seek generalizations of the finite-dimensional method to infinite-dimensional spaces via the spectral theory of linear operators, but I have no idea about the details.

So my question is whether there are any resources that deal systematically with infinite systems of difference equations. I would be grateful either for a resource that can be read also without a knowledge of advanced mathematics (that is something readable for non-mathematicians) or, alternatively, for a more advanced resource with a recommendation where to learn the prerequisities. My mathematical background is unfortunately quite basic (I am a computer science major): possibly relevant fields I have a background in are calculus, linear algebra, some fundamentals of modern analysis, difference/differential equations, and basics of functional analysis (however I have very poor knowledge of spectral theory).

I would appreciate any resources dealing with this issue, as well as any recommendations about which parts of mathematics I am supposed to learn (preferably with some recommended resources). I am willing to invest a considerable amount of time into the study, but I would like to follow some recommended path that leads to the goal. In the case that the theory of infinite systems of differential equations is similar, I would appreciate also resources dealing with them.

EDIT: an example of such system: $$x_{n+1} = A \cdot x_n,$$ where $x_n$ are in $\ell^{\infty}$, $x_0$ is given (for instance, let $x_0 = (1,0,1,0,1\ldots)$), and $A$ is a linear operator that can be viewed as an infinite matrix (for instance) $$A = \left(\begin{array}{ccccc}0 & 2 & 0 & 0 & \ldots \\ 2 & 0 & 1 & 0 & \ldots \\ 0 & 1 & 0 & 2 & \ldots \\ 0 & 0 & 2 & 0 & \ldots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right).$$ I have not been very careful, if this particular system makes sense and is well-behaved, but the form of my systems should be clear. $A$ is always a bounded operator on $\ell^{\infty}$ that can be (if needed) viewed as an infinite matrix that is $k$-diagonal for some fixed (relative to the system) constant $k$. However, these are details. I am not asking for a detailed solution, but mainly for a resource that treats such (or similar) systems and that will enable me to work out the details by myself.

ONE MORE EDIT: The matrix from the above example is symmetric by mistake. My matrices are not symmetric nor Hermitian in general (the operator is not necessarily self-adjoint).

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  • $\begingroup$ Can you give an example of such a system? Even a toy system would help clarify the concepts a bit... $\endgroup$ Commented Jul 21, 2013 at 17:40
  • $\begingroup$ @StevenStadnicki Thank you for your comment, I have edited the question with an example. $\endgroup$
    – 042
    Commented Jul 22, 2013 at 6:40
  • $\begingroup$ Can you give an example of a problem that has motivated you to consider systems of this form? $\endgroup$ Commented Jul 27, 2013 at 17:19

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So generalistic your question is the answer can be only general, some major research and application fields are

  • Z-transformation is used on such systems in the context of time discrete control systems. Hence you will find concepts within the field of cybernetics, systems' theory and control therory (input/output, impulse/response, filters).
  • In systems' theory regarding discrete systems on arbitrary sets; see for instance here
  • Particularly in systems' theory when some time-continuous systems which have delay properties, could be reformed in infinite difference systems as well.
  • In quantum mechanics such as discrete Schrödinger equations with infinity of solutions see for instance here
  • In synergetics for some spacial pattern formation the nonlinear version of infinite difference equations can be deducted by discretizing the spacial modes.

In general when discrete systems can be set up in terms of infinite number of state variables or modes, or continuous systems scattered in such discrete systems, then you will have areas where you can learn from for such equations.

It would be really far too wide to raise here references and more details; the list is really long but what may help you is to use google by highlighting explicitly the terms solution of systems of "infinite difference equation" (do use the quotes). Further the bold mark ups above will give you a focus on field of research where you can further dig into.

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