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I am studying an article in which convolution is used very much and it is used between all sorts of objects: functions, distributions and measures. I know that convolving a $L^1$ function (for example) with a regularizing kernel we get a smooth function. That's how we prove the density of the space of functions with compact support in $L^1$.

But how should I think of the convolution of a distribution with a regularizing kernel? Or the convolution of a measure with a regularizing kernel?

I haven't found any good and short introduction on the subject. If you have in mind some documents/books which provide a good introduction to the theory of convolution of distributions and measures, they are welcome. Thank you.

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Do you mean "probability distribution", "distribution" in the sense of "generalized function"? –  Michael Hardy Mar 4 '12 at 1:10
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2 Answers

up vote 1 down vote accepted

Here's a related question on mathoverflow: http://mathoverflow.net/questions/5892/what-is-convolution-intuitively

Here's what I wrote there:

begin quote

Among things that it's good to know about convolution is that the identity element for convolution is the Dirac delta function $δ$.

Another is that if you convolve a function $f$ with $δ'$, the derivative of the delta function, you get $f'$. Since convolution is associative, that implies that $f'∗g=f∗g'$.

Another is that often the convolution of two functions is as well-behaved as the better-behaved one of the two. If you convolve something with a smooth function, you get a smooth function; if you convolve something with a polynomial, you get a polynomial. In other words, many classes of "well-behaved" functions are ideals in a ring whose multiplication is convolution.

So if you convolve $f$ with a smooth approximation to Dirac's delta function, you get a smooth approximation to $f$. Thinking about why that works can probably shed a lot of intuitive light on the nature of convolution.

end quote

I'm guessing that by a regularizing kernel you mean a smooth approximation to the delta function. If you convolve a measure with one of those, you get another measure that is close to it. Smoothness of the kernel, with some additional hypotheses about the structure of stuff involved, will mean the density of the resulting measure will be smooth.

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Thank you for your answer. –  Beni Bogosel Mar 4 '12 at 12:20
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Concerning distributions perhaps that Zemanian's cheap book "Distribution Theory and Transform Analysis" could help.

Convolution with distributions is powerful since :

  • translation : $\delta(x-a) * T(x)=T(x-a)$
    (and unit of convolution considering the case $a=0$ !)
  • derivation : $\delta(x)' * T(x)= T'(x)$ (so that a differential equation may be rewritten as a convolution with a sum of $\delta$ distributions and derivatives justifying Heaviside's 'Symbolic Calculus')
  • integration : $H(x) * T(x)$ is a primitive of $T(x)$ with $H$ the Heaviside step function
  • ...

    Fourier transforms become rather interesting in distribution theory!

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Thank you for your answer. –  Beni Bogosel Mar 4 '12 at 12:20
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