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 : [itex]\frac{d^{\nu}}{dx^{\nu}}[/itex] or [itex]D^\nu[/itex] where the degree of derivation [itex]\nu[/itex] 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 [itex]\frac{d^{-\nu}}{dx^{-\nu}}[/itex] or [itex]D^{-\nu}[/itex] and the n-fold integral [itex]D^{-n}[/itex] in case of integer [itex]n[/itex] instead of real [itex]\nu[/itex]
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.