Inverse schwartz-distribution for convolution operation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:08:30Z http://mathoverflow.net/feeds/question/120975 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120975/inverse-schwartz-distribution-for-convolution-operation Inverse schwartz-distribution for convolution operation Unag 2013-02-06T14:36:35Z 2013-02-06T19:38:28Z <p>I note here $\mathcal{D}'$ the space of all distributions and $\mathcal{S}'$ the space of tempered distributions, I am considering the following question:</p> <p>Let $u \in \mathcal{D}'$ or $\mathcal{S}'$, I want to know general conditions such that we know there exists an inverse of $u$ for the convolution operation, meaning a distribution $v$ such that $u*v$ and $v*u$ can be defined and: $$u*v = v*u = \delta$$ </p> <p>When does a solution exist? When is that solution unique and can we describe all the solutions when it is not? </p> <p>Does it change the problem to only consider right or left-inverse of $u$?</p> <p>Thanks in advance.</p> http://mathoverflow.net/questions/120975/inverse-schwartz-distribution-for-convolution-operation/120986#120986 Answer by Liviu Nicolaescu for Inverse schwartz-distribution for convolution operation Liviu Nicolaescu 2013-02-06T16:26:23Z 2013-02-06T16:26:23Z <p>Your question is related to the famous and notoriously difficult <em>division problem</em>. If $u\in\mathscr{S}'$, and $\hat{u}$ is its Fourier transform, you ask when it is possible to define $\frac{1}{\hat{u}}$. Checl L. Schwartz's book <em>Theorie des Distributions</em>, Chapter V, Sections 4 and 5.</p> http://mathoverflow.net/questions/120975/inverse-schwartz-distribution-for-convolution-operation/121009#121009 Answer by Sönke Hansen for Inverse schwartz-distribution for convolution operation Sönke Hansen 2013-02-06T19:38:28Z 2013-02-06T19:38:28Z <p>The problem of division, for <code>$u$</code> having compact support, was solved by L. Ehrenpreis in the 1950's. The equation <code>$u*v=\delta$</code> is solvable if and only if the Fourier transform of <code>$u$</code> is slowly decreasing, which means that an estimate <code>$$\sup_{|\eta|&lt; \log(e+|\xi|)} |\hat u(\xi+\eta)|\geq C(1+|\xi|)^{-N}$$</code> holds. See, for example Theorem 16.5.22 in volume II of Hörmander's treatise on the Analysis of LPDO.</p>