# Space of Tikhonov regularization of an Ill poised problems.

This question is motivated by the answer given in the following link:

What space to use?

My understanding from the above answer is that given a problem and the properties we want the solution to have will determine which spaces we work in.

Now I am trying to the categorize the solution of a mathematical problem by "space wise". Is it possible? As an example, the Tikhonov regularization to solve an Ill poised problem is very well known. Now, can we know which one the one of the following is appropriate.

1) Is it a valid question to ask in which space we are working while doing Tikhonov regularization? If so which space is it?

2)The question (1) is not clear question because the space depends on the problem on which Tikhonov regularization is applied.

Is either (1) or (2) has a meaning in mathematics?

$$A:H^s\rightarrow H^{s+p},\quad p>0$$ This tells us that $A$ is "smoothing" in the sense that it makes functions which are only "$s$ times differentiable (weakly)" to functions which have "$p$ more derivatives". Such an operator cannot have a bounded inverse, i.e. the problem $$Ax = b$$ is ill-posed. This is essentially because the smoothing property of $A$ is killing off frequencies that we will never stably recover. Tikhonov regularization is a way to guarantee that we don't try to invert things that have these high frequencies present, hence eliminating the unboundedness issue and making the problem well-posed again.