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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?

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The typical convenient choice of spaces for Tikhonov regularization are the Sobolev spaces. The reason for this is that the Sobolev spaces characterize degrees of regularity, and as such they can be used to characterize the degree of ill-posedness of a particular equation. For example, if we have a linear operator

$$ A:H^s\rightarrow H^{s+p},\quad p>0 $$ 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.

To answer your question `how do I choose the right space before starting', the answer is usually you don't: you try a space, see what you can prove, then make changes as results allow. Sobolev spaces are a great first choice because they interact well with Fourier analysis, which in turn is the correct tool to analyze linear problems.

This theory is deep and technical, I suggest looking into this book by Tikhonov, and this book by Engl.

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  • $\begingroup$ Thank you so much. It clearly makes sense. Is Tikhonov regularization possible in Polish space? Should I ask as another question? $\endgroup$
    – Creator
    May 8 '15 at 1:25
  • $\begingroup$ I don't see any reason why you would need that much abstraction. Most ill-posed problems have very concrete formulations in terms of integral operators or solutions of differential equations, so Sobolev spaces are usually the correct tool. $\endgroup$
    – icurays1
    May 8 '15 at 1:32

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