The infinite-dimensional limit of sequence of solutions of linear equations when the number of equations goes to infinity

Solution 1:

I'm not sure what exactly you are looking for but if $\|C\|<+\infty$, $\sup_k \|C_k^{-1}\|<+\infty$, and $y\in\ell^2\cap C\ell^2$, then the convergence holds and, moreover, $\theta_k$ converge to the (unique) solution $\theta$ of $C\theta=y$ in $\ell^2$. The reason is very simple. If $P_k$ is the orthogonal projection to the first $k$ coordinates, then $C_k$ is the non-trivial part of $P_kCP_k$ so if $\theta_k$ is extended by zeroes, we have $P_kCP_k\theta_k=P_ky$. On the other hand, $|P_kCP_k\theta-P_ky|\le \|C\|\cdot|P_k\theta-\theta|$. Thus, $|\theta_k-P_k\theta|\le\|C_k^{-1}\|\cdot\|C\|\cdot |P_k\theta-\theta_k|\to 0$ as $k\to\infty$.

Solution 2:

This is closer to a comment than an answer, but it would be too long for a comment. Basically a few thougths that might or might not help. (Just a things this reminded me of, not necessarily helpfull for your problem.)

Let us rewrite your equation as $y^k=C_k\theta^k$. If I consider $y^k$ and $\theta^k$ as infinite sequences (I add zeroes), then I have

$$z^k=C\theta^k,$$

where the vector $z^k$ has the same values as $y^k$ on the first $k$ coordinates.

If we additionaly assume that $y^k\to y$ (which implies $z^k\to y$) and $\theta^k\to\theta$ pointwise and that the operator corresponding to $C$ is in some sense continuous, then we get

$$y=C\theta.$$

Hence a necessary condition is that $y$ is in the range of $C$.

My guess is that this condition should be sufficient for the continuity of $C$:

$$\|C\| = \sup_n \sum_k |c_{nk}| < \infty$$

I am not sure that this really helps but the last equation $$y=C\theta$$ is related to summability theory. In fact, matrix methods studied in summability theory are defined in the way that you transform a sequence using a given matrix and then take a limit.

For overview of some known results you can consult e.g. Boos: Classical and modern methods in summability, you can find also some notes here: http://thales.doa.fmph.uniba.sk/sleziak/texty/rozne/trf/boos/

From a functional analytic viewpoint: Morrison's Functional analysis