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Re: Delta functions.
Posted:
Apr 21, 2013 12:20 PM


Waldek Hebisch schrieb: > > clicliclic@freenet.de wrote: > > > > clicliclic@freenet.de schrieb: > > > > > > As I usually succeed in consistently computing with Dirac delta's, I've > > > felt no need to look into research on this. And while I don't know if > > > such research exists, my gut feeling is that the delta's cannot be > > > tweaked to qualify as members of your "field". By the way, since FriCAS > > > is a strongly typed system, what type does it assign to delta(x) where x > > > is a (say) complex irrational number? > > > > > > > Oops, this was nonsense: for 'complex' read 'real'. The Dirac delta > > makes sense for real arguments only, albeit in as many dimensions as one > > likes: it lives in R^n only: The flexibility required of the test > > functions cannot be achieved on C. > > > > Martin. > > ATM there is no delta finction in FriCAS. The place to add it > is 'Expression' domain. More precisly (since 'Expression' > need a parameter) domains like 'Expression(Integer)', > 'Expression(Fraction(Integer))', 'Expression(AlgebraicNumber)'. > > Note that analytically, delta at a complex number makes perfect > sense if you chose apropriate theory of distributions. > Similarly, in some theories of distributions you can > freely multiply them. This is one of the reasons I want > algebraic theory: such theory can be much more flexible > than usual analytic approach. > > Also, in Maple 'Dirac(x)' does not mean delta at point 'x', > but inverse image of delat at 0 via 'x'. If 'x' is a one > variable in multivariate context you get Lebesqu'e measure > at appropriate hyperplane. Using this convention 'Dirac(3)' > is just 0. >
I am ill equipped to discuss alternative theories of distributions since I know only of one ...
For members of a "field", division would have to be allowed as well. I remember dimly having seen something somewhere about multiplication of delta's  I imagine other properties would have to go in a theory that allows to multiply delta functions. Since distributions figure most prominently in the context of differential equations and integral transforms, multiplication by differentiable functions and differentiation are more important. It is also important that what is implemented in a CAS corresponds with what is commonly used.
There is nothing at all unusual about assigning delta(3) = 0 since (delta, f) = 0 for any test function f that vanishes outside some small environment of x = 3. See e.g. Constantinescu.
What particular theory of distributions do you have in mind that lends meaning to the delta function at a complex number? Perhaps somebody might want to look it up.
Martin.



