Talk:Convolution power

Tidy up

The following things need to be addressed:

• The whole of this article depends on the definition of negative convolution power, which in turn depends on the inverse Fourier transform. It must be made abundantly clear that for the majority of functions, this does not exist. I'm sure similar limitations apply to the exponentiation and logarithm stuff.

err.. what do you mean by the 'majority of functions' ? If you mean that the cardinality of the set of all functions from R->R (or C->C) on which the inverse transform operator may be applied is less than the cardinality of the set all functions from R->R (C->C), well, that limitation applies to pretty much all the calculus/analysis articles on wikipedia. So, I suppose, it would be good to be abundantly clear what you mean by 'the majority of functions'. Perhaps the best way would be to give some examples of these convolution operators applied to some functions, and to give some other examples where the operators fail. mike40033 (talk) 09:13, 14 August 2008 (UTC)

• The notation is highly inconsistent. If $\ k$ is a constant, then $\ x^{*(k)}$, $\ x^{*(y)}$ and $\ k^{*(y)}$ are all defined differently.

Partially fixed.

• The series expansion stuff is woefully under-explained -- what precisely are the "replacement" rules? This is probably because it's of very limited applicability (how exactly does it hold for "e.g. Fourier series"?). It's really just a limited case of the convolution theorem, which is much more general. Therefore, I'm not quite sure what the point of this section is.

Removed.

• The series expansion relationship involving $e^{*(x)}=\sum _{k=0}^{\infty }{\frac {x^{*(k)}}{k!}}$ is just a repetition of the previous definition.

Removed.

• Need references for definitions of the following (Google, etc. brings up practically nothing):
  • convolution power for anything other than the natural numbers
  • convolution exponentiation and logarithm section - where else are these definitions used??

Oli Filth 18:48, 5 July 2007 (UTC)

Certainly a bit of material on convolution exponential and logarithmic functions is found in Daniel Stroock's book Probability Theory: An Analytic View, in the section on infinite divisibility. Michael Hardy 13:15, 19 July 2007 (UTC)

Central limit theorem

Shouldn't there be some mention of the Central limit theorem as an application of convolution powers? -- Spireguy (talk) 22:10, 15 September 2009 (UTC)

Yes. I will add something. Sławomir Biały (talk) 11:41, 18 September 2009 (UTC)

Convolution root

Convolution power 1/2 is convolution root; see for instance CONVOLUTION ROOTS OF RADIAL POSITIVE DEFINITE FUNCTIONS WITH COMPACT SUPPORT by WERNER EHM, TILMANN GNEITING, AND DONALD RICHARDS. Boris Tsirelson (talk) 05:30, 27 August 2019 (UTC)