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What is the symbol of an integral operator?

  1. Jun 26, 2012 #1
    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!
     
  2. jcsd
  3. Jun 26, 2012 #2
    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
     
  4. Jun 27, 2012 #3
    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.
     
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