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Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is an old theorem, from 1932(ish). I am interested in generalizations such as:

  • When can a nonnegative $L^1$ function approximated by nonnegative linear combinations of translations of $f\ge 0$?
  • What about convex combinations for approximating a density function?
  • Other spaces: What about $L^1(X)$ or $L^p(X)$ for $X\ne\mathbb{R}$, $p\ne1,2$? (This question is also related.)

I am looking for any modern reference covering these types of extensions, or other interesting extensions of this theorem.

Edit: I eventually found this tome, which covers Tauberian theory in great detail, but not necessarily the approximation problems above. At 500 pages, it will take me some time to go through it.

Korevaar, Jacob, Tauberian theory. A century of developments, Grundlehren der Mathematischen Wissenschaften 329. Berlin: Springer (ISBN 3-540-21058-X/hbk). xvi, 483 p. (2004). ZBL1056.40002.

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Let me record a couple of simple observations regarding question 1.

First, it is clear that the answer is "very rarely", since there are very simple obstructions. For example, the step function $\mathbb{1}_{(0,1)}$ cannot be approximated by positive linear combinations of translates of $\mathbb{1}_{(0;3)}$. Indeed, if such a translate equals 1 at some point of $(0,1)$, then it is identically $1$ either on $(-1;0)$ or on $(1,2)$. Hence, any positive linear combination that is $\geq3/4$ at some point of $(0,1)$ will be $\geq 3/8$ either at all points of $(-1,0)$, or at all points of $(1,2)$. Admittedly, the Fourier transform of $\mathbb{1}_{(0;3)}$ does have zeros, but this is not relevant - the same argument can be adapted e. g. to Gaussians.

Second, assuming without loss of generality that $\int f=\int g=1$, I can try to find a probability measure $\mu$ on $\mathbb{R}$ such that $$ f\star \mu = g. $$Passing to Fourier transforms, this gives $\hat{\mu}=\hat{g}/\hat{f}$. We are asking how to determine whether a given function is a Fourier transform of a probability measure, which is classical - the characterization is given by Bochner's theorem and a useful sufficient condition - by Polya's criterion, see e. g. Durrett, Thm 3.3.22. Note that, in particular, we must have $|\hat{\mu}(t)|\leq \hat{\mu}(0)=1$, so that if $|\hat{f}(t)|<|\hat{g}(t)|$ for some $t$, they we have no solutions.

I guess if I have a sequence of probability measures such that $f\star \mu_n \to g$ in $L_1$, then $\mu_n$ must be tight and I could pass to a weakly convergent subsequence, and the limit must satisfy the equation above. So, Bochner's theorem really gives an "if and only if" characterisation.

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  • $\begingroup$ Can you give some more thoughts on why this is "very rare"? The example with the step function seems rather particular, and it is not clear to me that this generalizes. $\endgroup$ – JohnA Jun 6 '19 at 14:52
  • $\begingroup$ In fact it follows from the second part of the post that the step function example generalises to any non-negative $f$ and $g(x)=a^{-1}f(ax)$ with $0<a<1$. Indeed, in that case it is not hard to see that there will be a point where $|\hat{g}|>|\hat{f}|$, but a Fourier transform of probability measure is $\leq1$ by absolute value. $\endgroup$ – Kostya_I Jun 6 '19 at 17:29
  • $\begingroup$ But, perhaps, even more serious issue is that Bochner's theorem requires $\hat{g}/\hat{f}$ to be positive definite, which is a restrictive condition, and I see no reason why a ratio of two positive definite functions should be positive definite. $\endgroup$ – Kostya_I Jun 6 '19 at 17:33
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Regarding your last question, there is a theorem due to Beurling [1] that states that is the Hausdorff dimension of the null set of the Fourier transform is small (in terms of the dimension of $\mathbb{R}^d$, then the theorem holds in $L^p$ for $p$ far from $1$ in terms of the said dimension. I would search for papers citing Beurling.

[1] Beurling, Arne, On a closure problem, Ark. Mat. 1, 301-303 (1951). ZBL0042.35402.

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