What is the symbol of an integral operator?

[Crot]

Could someone please give a definition what the symbol of an integral operator is?
Does it have asymptotic expansions as symbols of differential operators?

I know about symbols of differential operators but, shame on me, heard nothing about the symbol of an integral operator. Any help in this direction will be kindly appreciated!

 

[JJacquelin]

Hi !

I understand that you are aware of the symbol for the differential operator : $\frac{d^{\nu}}{dx^{\nu}}$ or $D^\nu$ where the degree of derivation $\nu$ can be a non-integer number as well, thanks to the Riemann-Liouville transform (i.e. the fractional calculus)
This is extended to the integral operator $\frac{d^{-\nu}}{dx^{-\nu}}$ or $D^{-\nu}$ and the n-fold integral $D^{-n}$ in case of integer $n$ instead of real $\nu$
So, I think that the symbol is common for derivation and integration.
One can find alternative notations (but not on common use) on section 5 of the paper "La dérivation fractionnaire" and a short bibliography is provided on last page. http://www.scribd.com/JJacquelin/documents

 

[Crot]

Thank you for the answer! But, it is not what I meant. I did not mean the notation but
the symbol of a differential operator which can be expressed as a polynomial. You know, when one works with pseudo-differential operators, for example, such kind of symbols appear in quantity. But, I heard nothing about the symbol of a integral operator in this context. I hope I did make myself clearer.