Decomposition of distributions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:44:02Z http://mathoverflow.net/feeds/question/79808 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79808/decomposition-of-distributions Decomposition of distributions Marc Palm 2011-11-02T06:46:24Z 2011-11-06T22:59:03Z <p>Can we write every (tempered) distribution $\psi$, say on $\mathbb{R}$, as the sum of two distributions </p> <p>$\psi = \psi_1 + \psi_2$</p> <p>such that $\psi_1$ and the Fourier transform of $\psi_2$ are actually measurable functions of moderate growth. If so, under which additional conditions are the choices $\psi_1$ and $\psi_2$ unique?</p> http://mathoverflow.net/questions/79808/decomposition-of-distributions/79839#79839 Answer by BR for Decomposition of distributions BR 2011-11-02T14:41:10Z 2011-11-06T22:59:03Z <p>The <a href="http://en.wikipedia.org/wiki/Dirac_comb" rel="nofollow">Dirac Comb</a>, an infinite sum of delta functions, is an example of a tempered distribution that cannot be thusly decomposed (its Fourier transform is another Dirac Comb). </p> <p>[Added:] There is a positive result in this direction that I (among others) only partly-remembered: Any distribution can be written as a locally finite sum of derivatives of continuous functions. If the distribution has finite order, then the sum is finite. See Rudin's Functional Analysis, Theorem 6.28.</p>